3. Green’s currents [04ZC]

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3. Green’s currents

In this section, we study in detail some existence and regularity theory of Green’s currents. Our main motivation for studying these arises from Section 2, where we see the Green’s currents appear as Dirac type singular solutions to the linearization of dimension reduced Calabi-Yau equation by the $S^{1}$ -symmetry. It is possible that this study will also have applications to other geometric problems, especially to those concerning adiabatic limits.

This Section is organized as follows. In Section 3.1 we recall the generalized geodesic normal coordinates for an an embedded submanifold. In Section 3.2 we discuss the definition, local existence and regularity properties of Green’s currents in the general Riemannian setting. In Section 3.3, we refine these results in the special case related to Kähler geometry. In Section 3.4, we will prove a global existence result which will be immediately used in Section 4.

3.1. Normal coordinates for an embedded submanifold

We start our discussion by introducing the basic notions of the normal exponential map and normal coordinates with respect to an embedded submanifold. This part seems to be standard in Riemannian geometry. For the consistence of the notations and the completeness of the paper, here we include detailed discussions and proofs.

Let $(Q,g)$ be an oriented Riemannian manifold of dimension $m$ and let $P\subset Q$ be a closed embedded oriented submanifold of codimension $k_{0}$ in $Q$ . In our later applications, we only need the case $k_{0}=3$ . Denote by ${N}$ the normal bundle of $P$ in $Q$ , equipped with the induced fiberwise Riemannian inner products. For any $p\in P$ , $N(p)$ denotes the fiber of $p$ in $N$ . The normal exponential map of $P$ in $Q$ is defined as

(3.1)

\Exp_{P}:{N}\rightarrow Q,\ (p,v)\mapsto\Exp_{p}(v),\ v\in N(p),

where $\Exp_{p}:T_{p}Q\to Q$ is the standard exponential map at $p\in Q$ . By standard implicit function theorem, it is straightforward that $\Exp_{P}:{N}\to Q$ is a diffeomorphism from some neighborhood of the zero section of ${N}$ to some tubular neighborhood of $P$ in $Q$ .

Now we define the normal coordinates. Fix $p\in P$ . First we choose local coordinates $\{x_{1}^{\prime},\ldots,x_{m-k_{0}}^{\prime}\}$ in a small neighborhood $U\subset P$ of $p$ such that

(3.2)

\langle\partial_{x_{i}^{\prime}},\partial_{x_{j}^{\prime}}\rangle=\delta_{ij},\ 1\leq i,j\leq m-k_{0},

at $p$ . We also assume that $\partial_{x_{1}^{\prime}}\wedge\cdots\wedge\partial_{x_{m-k_{0}}^{\prime}}$ is compatible with the orientation on $U$ . Next, we pick local orthonormal sections $\{e_{1},\ldots,e_{k_{0}}\}$ of the normal bundle $N$ such that

(3.3)

\langle e_{\alpha},e_{\beta}\rangle=\delta_{\alpha\beta},\ 1\leq\alpha,\beta\leq k_{0},

on $U$ . Again we assume that $e_{1}\wedge\cdots\wedge e_{k_{0}}$ is compatible with the orientation on $N$ , i.e. $e_{1}\wedge\cdots\wedge e_{k_{0}}\wedge\partial_{x_{1}^{\prime}}\wedge\cdots\partial_{x_{k-m_{0}}^{\prime}}$ is compatible with the orientation on $Q$ . Then we can find $\epsilon>0$ such that $\Exp_{p}$ restricts to a diffeomorphism from

(3.4)

\mathfrak{S}_{0}=\{(q,v)\in N|q\in U,\ |v|<\epsilon\}

to a neighborhood $\mathcal{U}$ of $p$ . In particular, $U=\mathcal{U}\cap P$ .

Definition 3.1 (Normal coordinates).

For any $(q,v)\in\mathfrak{S}_{0}$ with $v=\sum\limits_{\alpha=1}^{k_{0}}v_{\alpha}e_{\alpha}$ , the local normal coordinates are defined as follows

(3.5)

\displaystyle\begin{cases}x_{j}(\Exp_{P}(q,v))\equiv x_{j}^{\prime}(q),&1\leq j\leq m-k_{0},\\ y_{\beta}(\Exp_{P}(q,v))\equiv v_{\beta},&1\leq\beta\leq k_{0}.\end{cases}

By definition for each fixed point $(y_{1},\ldots,y_{k_{0}},x_{1},\ldots,x_{m-k_{0}})$ in $\mathcal{U}$ , the curve

(3.6)

\vartheta(t)\equiv(ty_{1},\ldots,ty_{k_{0}},x_{1},\ldots,x_{m-k_{0}}),

is a normal geodesic which is orthogonal to $U\subset P$ . Let

(3.7)

\partial_{y_{1}},\ldots,\partial_{y_{k_{0}}},\partial_{x_{1}},\ldots,\partial_{x_{m-k_{0}}}

be the induced coordinate vector fields. Then $\partial_{y_{\alpha}}|_{y=0}=e_{\alpha}$ when both are viewed as sections of $N$ over $U\subset P$ . For $1\leq i,j\leq m-k_{0}$ and $1\leq\alpha\leq k_{0}$ , we denote

(3.8)

g_{ij}\equiv\langle\partial_{x_{i}},\partial_{x_{j}}\rangle,\ g_{i\alpha}\equiv\langle\partial_{x_{i}},\partial_{y_{\alpha}}\rangle,\ g_{\alpha\beta}\equiv\langle\partial_{y_{\alpha}},\partial_{y_{\beta}}\rangle.

By definition, we have

(3.9)

g_{ij}(0,0)=\delta_{ij},\ g_{\alpha\beta}(x,0)=\delta_{\alpha\beta},\ g_{i\alpha}(x,0)=0,

for all $x\in U\subset P$ . The second fundamental form of the embedding $U\hookrightarrow\mathcal{U}$ can be written as $\IIs=\IIs_{ij}^{\alpha}\partial_{y_{\alpha}}\otimes(dx_{i}\otimes dx_{j})$ , where

(3.10)

\IIs^{\alpha}_{ij}\equiv\langle\nabla_{\partial_{x_{i}}}e_{\alpha},\partial_{x_{j}}\rangle

Denote by $\overrightarrow{H}=H^{\alpha}e_{\alpha}$ the mean curvature vector, then

(3.11)

H^{\alpha}\equiv g^{ij}\IIs^{\alpha}_{ij}.

In the above coordinates, we define the normal distance function

(3.12)

r\equiv|y|=\Big(\sum\limits_{\alpha=1}^{k_{0}}y_{\alpha}^{2}\Big)^{\frac{1}{2}}.

A straightforward extension of the usual Gauss Lemma gives the following and we omit the proof.

Lemma 3.2 (Generalized Gauss Lemma).

For any $p\in P$ , there is some sufficiently small neighborhood $U\subset P$ of $p$ and a tubular neighborhood $\mathcal{U}\subset Q$ with $U=\mathcal{U}\cap P$ such that the function $r$ defined by (3.12) satisfies the following properties:

(1)

$\nabla r=\partial_{r}$ holds in $\mathcal{U}\setminus U$ . In particular, $r$ is the normal distance function in $\mathcal{U}$ , i.e., $r(q)=d(q,P)$ for all $q\in\mathcal{U}$ .

(2)

$\partial_{r}$ is orthogonal to $\partial_{x_{i}}$ ’s in $\mathcal{U}\subset Q$ , and hence

(3.13)

\langle\partial_{x_{i}},r\partial_{r}\rangle=\langle\partial_{x_{i}},\sum_{\alpha=1}^{k_{0}}y_{\alpha}\partial_{y_{\alpha}}\rangle=\sum_{\alpha=1}^{k_{0}}y_{\alpha}g_{i\alpha}=0,\ 1\leq i\leq m-k_{0}.

For the convenience of later discussion, we introduce several notations concerning the normal regularity order near the submanifold $P$ . It will be frequently used throughout the paper.

Definition 3.3 (Normal regularity order).

Let $T(x,y)$ be a tensor locally defined in $\mathcal{U}$ which is $C^{\infty}$ on $\mathcal{U}\setminus U$ , then for a non-negative integer $k$ we say as $r\rightarrow 0$

(1)

$T(x,y)=O^{\prime}(r^{k})$ if for each $\epsilon>0$

(3.14)

\displaystyle\Big|\partial_{x}^{I}\partial_{y}^{J}T(x,y)\Big|=\begin{cases}O(r^{-\epsilon}),&|J|\leq k,\\ O(r^{k-|J|-\epsilon}),&|J|>k,\end{cases}

for all multi-indices $I$ and $J$ . In particular, if $T\in C^{\infty}(\mathcal{U})$ , then $T=O^{\prime}(r^{k})$ for all $k\in\mathbb{Z}$ .

(2)

$T(x,y)=r^{k}O^{\prime}(1)$ if $r^{-k}T(x,y)=O^{\prime}(1)$ .
(3)

$T(x,y)=\widetilde{O}(r^{k})$ if $T(x,y)\in C^{\infty}(\mathcal{U})$ and

(3.15) $T(x,y)=r^{k}O^{\prime}(1).$

In other words, $T(x,y)$ is smooth in $\mathcal{U}$ and has vanishing normal derivatives along $U$ up to order $k-1$ .

Notice that the defining condition does not depend on the choice of the local coordinates, since if we have another coordinate system $\{\tilde{y}_{1},\cdots,\tilde{y}_{k_{0}},\tilde{x}_{1},\cdots,\tilde{x}_{m-k_{0}}\}$ , then we have

(3.16)		$\displaystyle\partial_{\tilde{y}_{\alpha}}=\frac{\partial y_{\beta}}{\partial\tilde{y}_{\alpha}}\cdot\partial_{y_{\beta}}+\frac{\partial x_{l}}{\partial\tilde{y}_{\alpha}}\cdot\partial_{x_{l}},$
(3.17)		$\displaystyle\partial_{\tilde{x}_{j}}=\frac{\partial x_{l}}{\partial\tilde{x}_{j}}\cdot\partial_{x_{l}}+\frac{\partial y_{\alpha}}{\partial\tilde{x}_{j}}\cdot\partial_{y_{\alpha}},$

and $\frac{\partial y_{\alpha}}{\partial\tilde{x}_{j}}\in rO^{\prime}(1)$ . Similarly, we can also use any local coordinate system $\{y_{\alpha},x_{i}\}$ such that $y_{\alpha}=0$ along $P$ for $1\leq\alpha\leq k_{0}$ .

Lemma 3.4.

In the above normal coordinates, we have the following expansions of the metric tensor $g$ of $Q$ along the normal directions,

(3.18)	$\displaystyle g_{\alpha\beta}\|_{(x,y)}$	$\displaystyle=\delta_{\alpha\beta}-\frac{1}{3}\Rm_{\alpha\gamma\xi\beta}\Big\|_{(x,0)}y_{\gamma}y_{\xi}+\widetilde{O}(r^{3}),$
(3.19)	$\displaystyle g_{ij}\|_{(x,y)}$	$\displaystyle=g^{P}_{ij}(x)+2\IIs^{\alpha}_{ij}\Big\|_{(x,0)}y_{\alpha}-(\Rm_{i\gamma\xi j}+\langle\nabla_{\partial_{x_{i}}}\partial_{y_{\gamma}},\nabla_{\partial_{x_{j}}}\partial_{y_{\xi}}\rangle)\Big\|_{(x,0)}y_{\gamma}y_{\xi}+\widetilde{O}(r^{3}),$
(3.20)	$\displaystyle g_{i\alpha}\|_{(x,y)}$	$\displaystyle=\langle\nabla_{\partial_{x_{i}}}\partial_{y_{\gamma}},\partial_{y_{\alpha}}\rangle\Big\|_{(x,0)}y_{\gamma}-\frac{2}{3}\Rm_{i\gamma\xi\alpha}\Big\|_{(x,0)}y_{\gamma}y_{\xi}+\widetilde{O}(r^{3}),$

where $g^{P}=(g^{P}_{ij})$ denotes the restriction of the metric $g$ to $U\subset P$ , $\Rm$ denotes the Riemann curvature tensor of $g$ .

Proof.

The above expansions can be proved using the Jacobi fields. Fix a point $q=(0^{k_{0}},x_{1},\ldots,x_{m-k_{0}})\in U\subset P$ , we we choose a unit vector $v=\sum\limits_{\alpha=1}^{m-k_{0}}v_{\alpha}\partial_{y_{\alpha}}\in N(q)\cong\mathbb{R}^{k_{0}}$ with $|v|=1$ . Let $\vartheta$ be the following radial geodesic in $\mathcal{U}$ ,

(3.21)

\vartheta(t)=\Exp_{P}(q,tv)\equiv\Exp_{q}(tv)

such that $\vartheta^{\prime}(0)=v$ . In the normal coordinates, the geodesic $\vartheta$ can represented as $\vartheta(t)=(tv_{1},\ldots,tv_{k_{0}},x_{1},\ldots,x_{m-k_{0}})$ .

For each $1\leq\alpha\leq k_{0}$ and $1\leq i\leq m-k_{0}$ , we define the geodesic variations

(3.22)		$\displaystyle\sigma_{\alpha}(t,s)$	$\displaystyle\equiv((tv_{1},\ldots,t(v_{\alpha}+s),\ldots,tv_{k_{0}},x_{1},\ldots,x_{m-k_{0}}),$
(3.23)		$\displaystyle\sigma_{i}(t,s)$	$\displaystyle\equiv(tv_{1},\ldots,tv_{k_{0}},x_{1},\ldots,x_{i}+s,\ldots x_{m-k_{0}}).$

Then variation fields of $\sigma_{\alpha}(t,s)$ and $\sigma_{i}(t,s)$ give the following Jacobi fields along the radial geodesic $\vartheta(t)$ respectively:

(3.24)

\displaystyle\begin{cases}J_{\alpha}(t)=t\cdot\partial_{y_{\alpha}},&1\leq\alpha\leq k_{0}\\ J_{i}(t)=\partial_{x_{i}},&1\leq i\leq m-k_{0}.\end{cases}

By definition,

(3.25)

J_{\alpha}(0)=0,\ J_{i}(0)=\partial_{x_{i}}.

Taking first derivatives at $t=0$ ,

(3.26)

J_{\alpha}^{\prime}(0)=\partial_{y_{\alpha}},\ J_{i}^{\prime}(0)=v_{\alpha}\nabla_{\partial_{x_{i}}}\partial_{y_{\alpha}}.

Then applying the Jacobi equation along the geodesic $\vartheta$ ,

(3.27)

\displaystyle\begin{cases}J_{\alpha}^{\prime\prime}+\Rm(J_{\alpha},\vartheta^{\prime})\vartheta^{\prime}=0,\\ J_{i}^{\prime\prime}+\Rm(J_{i},\vartheta^{\prime})\vartheta^{\prime}=0,\\ \end{cases}

where $\Rm(X,Y)Z\equiv\nabla_{X}\nabla_{Y}Z-\nabla_{Y}\nabla_{X}Z-\nabla_{[X,Y]}Z$ denotes the Riemann curvature tensor of $g$ , so it follows that

(3.28)		$\displaystyle J^{\prime\prime}_{\alpha}(0)=0,\ J^{\prime\prime\prime}_{\alpha}(0)=-v_{\gamma}v_{\xi}\Rm(\partial_{y_{\alpha}},\partial_{y_{\gamma}})\partial_{y_{\xi}},$
(3.29)		$\displaystyle J^{\prime\prime}_{i}(0)=-v_{\gamma}v_{\xi}\Rm(\partial_{x_{i}},\partial_{y_{\gamma}})\partial_{y_{\xi}}.$

Therefore,

(3.30)	$\displaystyle g_{\alpha\beta}$	$\displaystyle=t^{-2}g(J_{\alpha},J_{\beta})=\delta_{\alpha\beta}-\frac{1}{3}\Rm_{\alpha\gamma\xi\beta}v_{\gamma}v_{\xi}t^{2}+\widetilde{O}(t^{3}),$
(3.31)	$\displaystyle g_{ij}$	$\displaystyle=g(J_{i},J_{j})=g^{P}_{ij}+2v_{\alpha}\IIs_{ij}^{\alpha}t-\Big(\Rm_{i\gamma\xi j}+\langle\nabla_{\partial_{x_{i}}}\partial_{y_{\gamma}},\nabla_{\partial_{x_{j}}}\partial_{y_{\xi}}\rangle\Big)v_{\gamma}v_{\xi}t^{2}+\widetilde{O}(t^{3}),$
(3.32)	$\displaystyle g_{i\alpha}$	$\displaystyle=t^{-1}g(J_{i},J_{\alpha})=\langle\nabla_{\partial_{x_{i}}}\partial_{y_{\gamma}},\partial_{y_{\alpha}}\rangle v_{\gamma}t-\frac{2}{3}\Rm_{i\gamma\xi\alpha}v_{\gamma}v_{\xi}t^{2}+\widetilde{O}(t^{3}).$

Let $y_{\alpha}=tv_{\alpha}$ , then we obtain the desired expansions. ∎

As a digression we briefly discuss the intrinsic meaning of the above expansion. The point is that locally the Riemannian metric $g$ is approximated by a Riemannian metric $g_{N}$ on the normal bundle $N$ up to the first order. Notice we have the natural projection $\pi:N\rightarrow P$ and $N$ is a Riemannian vector bundle together with an induced “normal” connection, given by the normal component of the Levi-Civita connection of $g$ . The latter hence gives rise to a distribution of horizontal subspaces at each point of $N$ , which in our coordinates is spanned by $\partial_{x_{i}}-A_{i\alpha\beta}y_{\beta}\partial_{y_{\alpha}}$ , where

(3.33)

A_{i\alpha\beta}(x)\equiv\langle\nabla_{\partial_{x_{i}}}\partial_{y_{\beta}},\partial_{y_{\alpha}}\rangle|_{y=0}

is a smooth function on $U$ . We define $g_{N}$ so that at each point of $N$ , the vertical and horizontal subspaces are orthogonal and on the vertical part is given by the bundle metric on $N$ , and on the horizontal part is given by the perturbation of the base metric $g_{P}$ using the second fundamental form. In this way we get a coordinate free description of the above expansion up to the first order.

We also define

(3.34)

A_{ij\alpha\beta}\equiv\frac{1}{2}(\partial_{x_{i}}A_{j\alpha\beta}-\partial_{x_{j}}A_{i\alpha\beta}).

Then the curvature of the normal connection is given by

(3.35)

\Omega_{ij\alpha\beta}\equiv A_{ij\alpha\beta}+\frac{1}{2}(A_{i\alpha\gamma}A_{j\gamma\beta}-A_{i\beta\gamma}A_{j\gamma\alpha}).

3.2. Green’s currents for Riemannian submanifolds

First we recall and introduce the basic terminology. Let $(Q,g)$ be an oriented Riemannian manifold of dimension $m$ . Denote by $\Omega^{l}_{0}(Q)$ the space of differential $l$ -forms with compact supports in $Q$ .

Definition 3.5 ( $k$ -current).

A $k$ -current on $Q$ is a linear functional $T:\Omega^{m-k}_{0}(Q)\to\mathbb{R}$ which is continuous in the sense of distributions, i.e. suppose $\chi_{j}\in\Omega_{0}^{m-k}(Q)$ is a sequence of differential forms with all derivatives uniformly converging to $0$ as $j\to\infty$ , then $\lim\limits_{j\to\infty}(T,\chi_{j})=0$ .

The notion of currents unifies the notion of differential forms and submanifolds. In particular, a locally integrable $k$ -form $\beta$ can be naturally viewed as a $k$ -current via the pairing

(3.36)

(\beta,\chi)\equiv\int_{Q}\beta\wedge\chi,\ \ \ \ \chi\in\Omega^{m-k}_{0}(Q),

and an oriented submanifold $P$ of co-dimension $k$ also defines a $k$ -current via

(3.37)

(\delta_{P},\chi)\equiv\int_{P}\chi,\ \ \ \ \chi\in\Omega^{m-k}_{0}(Q).

The usual exterior differential $d$ and the Hodge star operator $*$ on differential forms then naturally extend to currents. Given a $k$ -current $T$ and $\chi\in\Omega_{0}^{m-k}(Q)$ , then we define

(3.38)		$\displaystyle(dT,\chi)\equiv(-1)^{k+1}(T,d\chi),$
(3.39)		$\displaystyle(T,\chi)\equiv(-1)^{k(m-k)}(T,\chi).$

Let $d^{*}$ be the codifferential operator and denote by $\Delta\equiv dd^{*}+d^{*}d$ the Hodge Laplacian, then it follows that for every $k$ -current $T$ and $\chi\in\Omega_{0}^{m-k}(Q)$ ,

(3.40)		$\displaystyle(d^{}T,\chi)=(-1)^{k}(T,d^{}\chi),$
(3.41)		$\displaystyle(\Delta T,\chi)=(T,\Delta\chi).$

A $k$ -current $T$ is called harmonic if $\Delta T=0$ . It follows from the standard elliptic regularity theory that a harmonic $k$ -current can be represented by a smooth harmonic $k$ -form.

Now let $P\subset Q$ be a (not necessarily closed) embedded oriented submanifold. Although the following discussion applies to more general setting, for our purpose in the following we will only consider the case when $P$ is of co-dimension $3$ in $Q$ . The importance of the co-dimension $3$ case in our setting is related to the fact that there is a Hopf fibration $\mathbb{R}^{4}\rightarrow\mathbb{R}^{3}$ which is a singular $S^{1}$ fibration with a smooth total space and co-dimension discriminant locus on the base. The co-dimension 3 condition also appears in other geometric settings, for example, Hitchin’s theory of Gerbes [Hit01].

Definition 3.6 (Green’s current).

Suppose $P$ is of co-dimension 3 in $Q$ . A Green’s current $G_{P}$ for $P$ in $Q$ is a locally integrable $3$ -form which solves the following current equation on $Q$

(3.42)

\Delta G_{P}=2\pi\cdot\delta_{P}.

Example 3.7.

The above normalization constant is chosen such that in the case $Q\equiv\mathbb{R}^{3}$ and $P\equiv 0^{3}\in\mathbb{R}^{3}$ , then

(3.43)

G_{P}=\frac{1}{2|y|}dy_{1}\wedge dy_{2}\wedge dy_{3}

solves the current equation $\Delta_{0}G_{P}=2\pi\cdot\delta_{P}$ for the standard Hodge Laplacian $\Delta_{0}$ on $\mathbb{R}^{3}$ .

In particular $G_{P}$ is harmonic outside $P$ hence is smooth. Notice a Green’s current $G_{P}$ for $P$ is not unique, but it is unique up to the addition of a harmonic $3$ -form, so the singular behavior near $P$ does not depend on the particular choice of $G_{P}$ . Also it is clear that if $Q^{\prime}\subset Q$ is an open submanifold, then the restriction of $G_{P}$ to $Q^{\prime}$ is a Green’s current for $P^{\prime}=P\cap Q^{\prime}$ in $Q^{\prime}$ , so that we can study the regularity problem locally. Our goal in this subsection is to understand the local existence and regularity of $G_{P}$ via approximation by the standard model, which is the product space $\mathbb{R}^{3}\times\mathbb{R}^{n-3}$ .

To begin with, we have the following simple regularity result for $d(G_{P})$ .

Proposition 3.8.

Given a Green’s current $G_{P}$ , its differential $d(G_{P})$ extends to a smooth $4$ -form across $P$ .

Proof.

This is a local result so we can work with the geodesic ball $B_{r}(p)$ for any $p\in P$ such that $\overline{B_{r}(p)}\subset\subset Q$ and $\overline{B_{r}(p)}\cap P\subset\subset P$ . We will show that the $4$ -current $dG_{P}$ is a harmonic in $B_{r}(p)$ in the distributional sense. In fact, for any test form $\chi\in\Omega^{m-4}_{0}(B_{r}(p))$ we have

(3.44)

(\Delta(d(G_{P})),\chi)=(d\Delta G_{P},\chi)=(\Delta G_{P},d\chi)=(2\pi\delta_{P},d\chi)=2\pi\int_{P}d\chi=2\pi\int_{\partial B_{r}(p)\cap P}\chi=0.

Therefore, $d(G_{P})$ is a harmonic $4$ -current in $B_{r}(p)$ and hence it is smooth in $B_{r}(p)$ . ∎

In the rest of this section, we will frequently use the following notation.

Notation 3.9.

Given $k\in\mathbb{Z}_{+}$ , a capital Greek letter with index $k$ , $\Pi_{p}^{(k)}$ , always denotes a general local $p$ -form with $0\leq p\leq 3$ , which is of the form

(3.45)

\Pi_{p}^{(k)}\equiv P_{i,j_{1},\ldots,j_{p-1}}^{(k)}(x,y)(dy_{j_{1}}\wedge\ldots\wedge dy_{j_{p-1}})\wedge dx_{i}+Q_{j_{1},\ldots,j_{p}}^{(k)}(x,y)dy_{j_{1}}\wedge\ldots\wedge dy_{j_{p}},

where $P_{i,j_{1},\ldots,j_{p-1}}^{(k)}(x,y)$ and $Q_{j_{1},\ldots,j_{p}}^{(k)}(x,y)$ are homogeneous polynomial functions in $y$ of degree $k$ whoses coefficient functions are smooth on $U\subset P$ .

Remark 3.9.1.

Notice this expression depends on the choice of local coordinates, but under a change of coordinates, a $p$ -form $\Pi_{p}^{(k)}$ will still have such an expression, modulo a term which is of order $\widetilde{O}(r^{k+1})$ .

Now we are ready to state the first main theorem of this section, which gives a local existence for Green’s current, and its leading singular behavior.

Theorem 3.10.

If $P\subset Q$ is an embedded submanifold of co-dimension $3$ , then for any $p\in P$ , there is a neighborhood $\mathcal{U}$ of $p$ in $Q$ , and a Green’s current $G_{U}$ for $U=\mathcal{U}\cap P$ in $\mathcal{U}$ satisfying

(3.46)

\displaystyle\Delta G_{U}=2\pi\cdot\delta_{U}\ \ \ \ \text{in}\ \ \ \ \ \mathcal{U},

and the local expansion

	$\displaystyle G_{U}=$	$\displaystyle\frac{1}{2r}(1-\frac{H^{\alpha}y_{\alpha}}{2})dy_{1}\wedge dy_{2}\wedge dy_{3}+\frac{1}{2r}y_{\beta}A_{i\alpha\beta}dx_{i}\wedge dy_{\widehat{\alpha}}-\frac{1}{4}A_{ij\alpha\beta}r\cdot dy_{\widehat{\alpha\beta}}\wedge dx_{i}\wedge dx_{j}$
	$\displaystyle+$	$\displaystyle\frac{3}{16}(A_{i\alpha,\alpha+1}A_{j\alpha,\alpha+2}-A_{i\alpha,\alpha+2}A_{j\alpha,\alpha+1})d(ry_{\alpha})\wedge dx_{i}\wedge dx_{j}+r^{-3}\Pi_{3}^{(4)}+O^{\prime}(r^{2}),$

where $\overrightarrow{H}=H^{\alpha}e_{\alpha}$ is the mean curvature of $P\subset Q$ and the $3$ -form $\Pi_{3}^{(4)}$ is defined in (3.45).

Here and in the following we use the notation that for $\alpha\in\{1,2,3\}$ , $\widehat{\alpha}=(\alpha+1,\alpha+2)$ (with the convention $3+1=1$ ), $dy_{\alpha\beta}\equiv dy_{\alpha}\wedge dy_{\beta}$ , and that

(3.48)

\displaystyle dy_{\widehat{\alpha\beta}}\equiv\begin{cases}dy_{\alpha+2},&\beta=\alpha+1\\ -dy_{\alpha+1},&\beta=\alpha+2\\ 0,&\beta=\alpha.\end{cases}

Remark 3.10.1.

By the above discussion, if $G_{P}$ is any Green’s current for $P$ in $Q$ , then locally near $p\in P$ , $G_{P}$ will also have an expansion of the form (3.10).

Remark 3.10.2.

At a given point $p\in U$ , we can always choose a special frame $\{e_{\alpha}\}$ such that $A_{i\alpha\beta}=0$ at $p$ . On the other hand, since the singular behavior of $G_{U}$ does not depend on the choice of coordinates, one sees that in general we need the second term in the expansion.

Remark 3.10.3.

By Proposition 3.8, $dG_{U}$ is smooth. This is compatible with the above expansion. For example, from the expansion we see the leading term involving forms of type $dx_{i}\wedge dx_{j}\wedge dy_{\beta}\wedge dy_{\gamma}$ is given by

(3.49)

(\frac{1}{2r}y_{\beta}A_{ij\alpha\beta}dy_{\hat{\alpha}}-\frac{1}{4}A_{ij\alpha\beta}dr\wedge dy_{\widehat{\alpha\beta}})\wedge dx_{i}\wedge dx_{j}.

Elementary calculation shows this vanishes.

Remark 3.10.4.

In the proof we shall not keep track of the explicit form of $\Pi_{3}^{(4)}$ because it is not needed in our applications. However, it is possible to obtain the precise expression with more work. Given the above expansion, there are also some constraint for $\Pi_{3}^{(4)}$ following from the fact that $d(G_{U})$ is smooth by Proposition 3.8.

Before starting the proof of Theorem 3.10, we need some preparations. For the convenience of our calculations, we introduce three $1$ -forms

(3.50)

\eta_{\alpha}\equiv dy_{\alpha}+p_{i\alpha}dx_{i}

such that

(3.51)

\langle\eta_{\alpha},dx_{j}\rangle=0

for all $j$ and $\alpha$ at all points of $\mathcal{U}$ . Then the linear span of the $\eta_{\alpha}$ ’s is orthogonal to the linear span of the $dx_{i}$ ’s.

The lemma below is s crucial in the proof of Theorem 3.10.

Lemma 3.11.

For any $1\leq k\leq m-3$ and $1\leq\alpha\leq 3$ ,

(3.52)

p_{k\alpha}=g_{k\alpha}+\widetilde{O}(r^{3}).

Proof.

We write the full matrix expression of the metric $g$ as

(3.53)

g=\left[{\begin{array}[]{cc}g_{\alpha\beta}&0\\ 0&g_{ij}\\ \end{array}}\right]+\left[{\begin{array}[]{cc}0&S\\ S^{t}&0\\ \end{array}}\right],

where $S=(g_{\alpha i})=\widetilde{O}(r)$ . We denote by $(h_{\alpha\beta})$ and $(h_{ij})$ the inverse matrix of $(g_{\alpha\beta})$ and $(g_{ij})$ respectively. Then by elementary consideration

(3.74)	$\displaystyle g^{-1}$	$\displaystyle=$	$\displaystyle\left[{\begin{array}[]{cc}h_{\alpha\beta}&0\\ 0&h_{ij}\\ \end{array}}\right]-\left[{\begin{array}[]{cc}h_{\alpha\beta}&0\\ 0&h_{ij}\\ \end{array}}\right]\left[{\begin{array}[]{cc}0&S\\ S^{t}&0\\ \end{array}}\right]\left[{\begin{array}[]{cc}h_{\alpha\beta}&0\\ 0&h_{ij}\\ \end{array}}\right]$
		$\displaystyle+$	$\displaystyle\left[{\begin{array}[]{cc}h_{\alpha\beta}&0\\ 0&h_{ij}\\ \end{array}}\right]\left[{\begin{array}[]{cc}0&S\\ S^{t}&0\\ \end{array}}\right]\left[{\begin{array}[]{cc}h_{\alpha\beta}&0\\ 0&h_{ij}\\ \end{array}}\right]\left[{\begin{array}[]{cc}0&S\\ S^{t}&0\\ \end{array}}\right]\left[{\begin{array}[]{cc}h_{\alpha\beta}&0\\ 0&h_{ij}\\ \end{array}}\right]$
		$\displaystyle+$	$\displaystyle\widetilde{O}(r^{3}).$

Notice the third term does not have off-diagonal contributions, so the inverse matrix $g^{-1}=(g^{IJ})$ satisfies

(3.75)	$\displaystyle g^{i\alpha}$	$\displaystyle=-h_{ij}g_{j\beta}h_{\beta\alpha}+\widetilde{O}(r^{3})$
(3.76)	$\displaystyle g^{ij}$	$\displaystyle=h_{ij}+\widetilde{O}(r^{2}),$
(3.77)	$\displaystyle g^{\alpha\beta}$	$\displaystyle=h_{\alpha\beta}+\widetilde{O}(r^{2})=\delta_{\alpha\beta}+\widetilde{O}(r^{2}),$

where we used Lemma 3.4. The definition of $\eta_{\alpha}$ requires $\langle\eta_{\alpha},dx_{j}\rangle=0$ , which implies that

(3.78)

p_{i\alpha}g^{ij}+g^{j\alpha}=0.

Let $(\hat{g}_{ij})$ be the inverse of the matrix $(g^{ij})$ with $1\leq i,j\leq m-3$ such that $g^{ij}\hat{g}_{jk}=\delta_{ik}$ . Multiplying by $\hat{g}_{jk}$ , we have

(3.79)

p_{i\alpha}g^{ij}\hat{g}_{jk}+g^{j\alpha}\hat{g}_{jk}=0,

and hence

(3.80)

p_{k\alpha}=-g^{j\alpha}\hat{g}_{jk}.

We claim that for any $1\leq i,j\leq m-3$ ,

(3.81)

\hat{g}_{ij}-g_{ij}=\widetilde{O}(r^{2}).

In fact, since

(3.82)

g^{ij}g_{jk}+g^{i\alpha}g_{\alpha k}=\delta_{ik}.

Multipling by the inverse of the submatrix $(g^{ij})$ ,

(3.83)

\hat{g}_{li}(g^{ij}g_{jk}+g^{i\alpha}g_{\alpha k})=\hat{g}_{lk}.

So this implies that

(3.84)

\hat{g}_{lk}-g_{lk}=\hat{g}_{li}g^{i\alpha}g_{\alpha k}=\widetilde{O}(r^{2}).

Therefore, combining (3.75),(3.77), (3.80) and (3.84), we obtain

(3.85)	$\displaystyle p_{k\alpha}$	$\displaystyle=$	$\displaystyle-(g_{kj}+\widetilde{O}(r^{2}))g^{j\alpha}$
		$\displaystyle=$	$\displaystyle-g_{kj}g^{j\alpha}+\widetilde{O}(r^{3})$
		$\displaystyle=$	$\displaystyle h_{\alpha\beta}g_{k\beta}+\widetilde{O}(r^{3})$
		$\displaystyle=$	$\displaystyle g_{k\alpha}+\widetilde{O}(r^{3}).$

∎

The following symmetry property of $p_{i\alpha}$ will be frequently used in our later calculations. By Lemma 3.4 and Lemma 3.11, we may write

(3.86)

p_{i\alpha}=A_{i\alpha\beta}y_{\beta}+\frac{1}{2}B_{i\alpha\beta\gamma}y_{\beta}y_{\gamma}+\widetilde{O}(r^{3}).

Here the connection term $A_{i\alpha\beta}\equiv\langle\nabla_{\partial_{x_{i}}}\partial_{y_{\beta}},y_{\alpha}\rangle|_{(x,0)}$ is skew-symmetric in $\alpha$ , $\beta$ , and the curvature term $B_{i\alpha\beta\gamma}\equiv-\frac{4}{3}R_{i\beta\gamma\alpha}$ is symmetric in $\beta$ , $\gamma$ .

Lemma 3.12.

For every $1\leq\alpha,\beta\leq 3$ ,

(3.87)

\langle\eta_{\alpha},\eta_{\beta}\rangle=\delta_{\alpha\beta}+\widetilde{O}(r^{2}).

Proof.

By definition

(3.88)

\langle\eta_{\alpha},\eta_{\beta}\rangle=\langle dy_{\alpha}+p_{i\alpha}dx_{i},dy_{\beta}+p_{j\beta}dx_{j}\rangle.

By (3.77) we get

(3.89)

\langle dy_{\alpha},dy_{\beta}\rangle=\delta_{\alpha\beta}+\widetilde{O}(r^{2}).

Also we have $p_{i\alpha}=\widetilde{O}(r)$ and $\langle dx_{i},dy_{\beta}\rangle=\widetilde{O}(r)$ for all $i$ and $\alpha$ . The conclusion then follows. ∎

Using the above differential forms $\eta_{\alpha}$ ’s, we can decompose the volume form $\dvol_{g}$ in the horizontal and vertical directions, which will substantially simplify the computations regarding the Hodge Laplacian. The volume form of $g$ is given by

(3.90)

\dvol_{g}=\sqrt{\det(g)}\cdot dy_{1}\wedge dy_{2}\wedge dy_{3}\wedge dx_{1}\wedge\cdots\wedge dx_{m-3},

where we have used the orientation fixed above. We define the normal and tangential volume forms by

(3.91)

\displaystyle\begin{cases}{\dvol}_{N}\equiv\eta_{1}\wedge\eta_{2}\wedge\eta_{3}\\ {\dvol}_{T}\equiv\sqrt{\det(g_{ij}^{P})}\cdot dx_{1}\wedge\cdots\wedge dx_{m-3}.\end{cases}

By the expansion formula (3.86), the normal volume form $\dvol_{N}$ has the following expansion,

	$\displaystyle{\dvol}_{N}=dy_{1}\wedge dy_{2}\wedge dy_{3}+(A_{i\alpha\beta}y_{\beta}+\frac{1}{2}B_{i\alpha\beta\gamma}y_{\beta}y_{\gamma})dx_{i}\wedge dy_{\widehat{\alpha}}$
(3.92)		$\displaystyle+A_{i,\alpha+1,\beta}A_{j,\alpha+2,\gamma}y_{\beta}y_{\gamma}dy_{\alpha}\wedge dx_{i}\wedge dx_{j}+\widetilde{O}(r^{3}).$

In addition, by the definition of $\eta_{\alpha}$ ’s, it holds that for each $1\leq\alpha\leq 3$ ,

(3.93)

\eta_{\alpha}\wedge\dvol_{T}=dy_{\alpha}\wedge\dvol_{T},

and hence

(3.94)

\dvol_{N}\wedge\dvol_{T}=dy_{1}\wedge dy_{2}\wedge dy_{3}\wedge\dvol_{T}=\frac{\sqrt{\det(g_{ij}^{P})}}{\sqrt{\det(g)}}\cdot\dvol_{g}.

Lemma 3.13.

Denote by $*:\Omega^{k}(Q)\to\Omega^{m-k}(Q)$ the Hodge $*$ operator, then we have the following:

(1)

(3.95) $*\dvol_{T}=(-1)^{m+1}\dvol_{N}\cdot(1-H^{\alpha}y_{\alpha}+\widetilde{O}(r^{2})),$

(2)

For any $\alpha\in\{1,2,3\}$ ,

(3.96)

*(\eta_{\alpha}\wedge\dvol_{T})=\lambda_{1}\eta_{\widehat{\alpha}}+\lambda_{2}\eta_{\widehat{\alpha+1}}+\lambda_{3}\eta_{\widehat{\alpha+2}},

where $\lambda_{1}=1-H^{\alpha}y_{\alpha}+\widetilde{O}(r^{2})$ , $\lambda_{2}=\widetilde{O}(r^{2})$ , $\lambda_{3}=\widetilde{O}(r^{2})$ .

Proof.

First, we prove Item (1). By (3.51) we have

(3.97)

(-1)^{m+1}*\dvol_{T}=\lambda\cdot\dvol_{N}

for a function $\lambda>0$ . The function $\lambda$ is given by

(3.98)

\lambda=\frac{|\dvol_{T}|^{2}\sqrt{\det(g)}}{\sqrt{\det(g_{ij}^{P})}}=\sqrt{\det(g)}\det(g^{ij})\sqrt{\det(g_{ij}^{P})}.

Now we compute the expansion of $\lambda$ . Applying the expansions of $g_{ij}$ , $g_{\alpha\beta}$ and $g_{i\alpha}$ in Lemma 3.4, one can directly obtain the following,

(3.99)	$\displaystyle\det(g)$	$\displaystyle=\det(g_{\alpha\beta})\cdot\det(g_{ij})+\widetilde{O}(r^{2}),$
(3.100)	$\displaystyle\det(g_{\alpha\beta})$	$\displaystyle=1+\widetilde{O}(r^{2}),$
(3.101)	$\displaystyle\det(g_{ij})$	$\displaystyle=\det(g_{ij}^{P})\cdot(1+2H^{\alpha}y_{\alpha})+\widetilde{O}(r^{2}).$

Plugging (3.100) and (3.101) into (3.99),

(3.102)

\det(g)=\det(g_{ij}^{P})\cdot(1+2H^{\alpha}y_{\alpha})+\widetilde{O}(r^{2}).

Let $(h_{ij})$ be the inverse of the matrix $(g_{ij})$ . Since $g^{ij}=h_{ij}+\widetilde{O}(r^{2})$ by (3.76), so it follows that

(3.103)

\det(g^{ij})=\det(h_{ij})+\widetilde{O}(r^{2})=(\det(g_{ij}))^{-1}+\widetilde{O}(r^{2}).

Plugging (3.101) into the above,

(3.104)

\det(g^{ij})=\det(g_{ij}^{P})^{-1}\cdot(1-2H^{\alpha}y_{\alpha})+\widetilde{O}(r^{2}).

Therefore, substituting (3.102) and (3.104) into (3.98),

(3.105)

\lambda=1-H^{\alpha}y_{\alpha}+\widetilde{O}(r^{2}),

which completes the proof of Item (1).

Now we prove Item (2). For each $\alpha\in\{1,2,3\}$ , we can write

(3.106)

*(\eta_{\alpha}\wedge\dvol_{T})=\lambda_{1}\cdot\eta_{\widehat{\alpha}}+\lambda_{2}\cdot\eta_{\widehat{\alpha+1}}+\lambda_{3}\cdot\eta_{\widehat{\alpha+2}}.

Taking point-wise wedge product with $\eta_{\alpha}\wedge\dvol_{T}$ , and noticing $\eta_{\widehat{\alpha+1}}\wedge\eta_{\alpha}$ , $\eta_{\widehat{\alpha+2}}\wedge\eta_{\alpha}$ are both zero, then we obtain

(3.107)

\lambda_{1}\cdot\eta_{\widehat{\alpha}}\wedge\eta_{\alpha}\wedge\dvol_{T}=(\eta_{\alpha}\wedge\dvol_{T})\wedge*(\eta_{\alpha}\wedge\dvol_{T}).

(3.108)

\displaystyle\lambda_{1}\cdot\dvol_{N}\wedge\dvol_{T}=|\eta_{\alpha}\wedge\dvol_{T}|^{2}\dvol_{g}

Therefore, by (3.51) and (3.87) we get

(3.109)

\lambda_{1}=\frac{|\eta_{\alpha}\wedge\dvol_{T}|^{2}\sqrt{\det(g)}}{\sqrt{\det(g_{ij}^{P})}}=1-H^{\alpha}y_{\alpha}+\widetilde{O}(r^{2}),

Similarly taking wedge product with $\eta_{\alpha+1}\wedge\dvol_{T}$ and $\eta_{\alpha+2}\wedge\dvol_{T}$ respectively, and again by (3.87) we obtain that

(3.110)

\lambda_{2}=\widetilde{O}(r^{2}),\lambda_{3}=\widetilde{O}(r^{2}).

These imply that

(3.111)

*(\eta_{\alpha}\wedge\dvol_{T})=\lambda_{1}\cdot\eta_{\widehat{\alpha}}+\lambda_{2}\cdot\eta_{\widehat{\alpha+1}}+\lambda_{3}\cdot\eta_{\widehat{\alpha+2}},

where $\lambda_{1}=1+H^{\alpha}y_{\alpha}+\widetilde{O}(r^{2})$ , $\lambda_{2}=\widetilde{O}(r^{2})$ and $\lambda_{3}=\widetilde{O}(r^{2})$ . ∎

Now we proceed to prove Theorem 3.10. This will be done in several steps.

Step 1. We start by defining a 3-form

(3.112)

\phi_{1}\equiv(-1)^{m+1}\frac{1}{2r}*\dvol_{T}.

By Item (1) of Lemma 3.13, immediately we have

(3.113)

\displaystyle\phi_{1}=\frac{\dvol_{N}}{2r}\cdot(1-H^{\alpha}y_{\alpha}+\widetilde{O}(r^{2})).

Then applying the expansion of $\dvol_{N}$ in (3.92), $\phi_{1}$ has a further expansion,

	$\displaystyle\phi_{1}$	$\displaystyle=\frac{1-H^{\alpha}y_{\alpha}}{2r}dy_{1}\wedge dy_{2}\wedge dy_{3}+\frac{1}{2r}A_{i\alpha\beta}y_{\beta}dx_{i}\wedge dy_{\widehat{\alpha}}$
		$\displaystyle+\frac{1}{2r}A_{i,\alpha+1,\beta}A_{j,\alpha+2,\gamma}y_{\beta}y_{\gamma}dy_{\alpha}\wedge dx_{i}\wedge dx_{j}+r^{-1}\Pi_{3}^{(2)}+O^{\prime}(r^{2})$
		$\displaystyle=\frac{1-H^{\alpha}y_{\alpha}}{2r}dy_{1}\wedge dy_{2}\wedge dy_{3}+\frac{1}{2r}A_{i\alpha\beta}y_{\beta}dx_{i}\wedge dy_{\hat{\alpha}}$
(3.114)			$\displaystyle+\frac{1}{4}(A_{i\alpha,\alpha+1}A_{j\alpha,\alpha+2}-A_{i\alpha,\alpha+2}A_{j\alpha,\alpha+1})y_{\alpha}\cdot dr\wedge dx_{i}\wedge dx_{j}+r^{-1}\Pi_{3}^{(2)}+O^{\prime}(r^{2}),$

where $\Pi_{3}^{(2)}$ is the $3$ -form introduced in Notation 3.9 and the last step can be achieved by applying the following lemma:

Lemma 3.14 (Rearrangement Lemma).

(3.115)

A_{i,\alpha+1,\beta}A_{j,\alpha+2,\gamma}y_{\beta}y_{\gamma}dy_{\alpha}\wedge dx_{i}\wedge dx_{j}=\frac{1}{2}(A_{i\alpha,\alpha+1}A_{j\alpha,\alpha+2}-A_{i\alpha,\alpha+2}A_{j\alpha,\alpha+1})y_{\alpha}\cdot rdr\wedge dx_{i}\wedge dx_{j}.

Proof.

First by writing out the terms and re-arranging the subscripts and using the skew symmetry of $A_{i\alpha\beta}$ we get

(3.116)		$\displaystyle A_{i,\alpha+1,\beta}A_{j,\alpha+2,\gamma}y_{\beta}y_{\gamma}dy_{\alpha}$
	$\displaystyle=$	$\displaystyle\Big(A_{i,\alpha+1,\alpha}A_{j,\alpha+2,\alpha}y_{\alpha}^{2}+A_{i,\alpha+1,\alpha}A_{j,\alpha+2,\alpha+1}y_{\alpha}y_{\alpha+1}$
		$\displaystyle+A_{i,\alpha+1,\alpha+2}A_{j,\alpha+2,\alpha}y_{\alpha}y_{\alpha+2}+A_{i,\alpha+1,\alpha+2}A_{j,\alpha+2,\alpha+1}y_{\alpha+1}y_{\alpha+2}\Big)dy_{\alpha}$
	$\displaystyle=$	$\displaystyle A_{i,\alpha,\alpha+1}A_{j,\alpha,\alpha+2}y_{\alpha}^{2}dy_{\alpha}-A_{i\alpha,\alpha+2}A_{j\alpha,\alpha+1}y_{\alpha}y_{\alpha+2}dy_{\alpha+2}$
		$\displaystyle-A_{i,\alpha,\alpha+2}A_{j,\alpha,\alpha+1}y_{\alpha}y_{\alpha+1}dy_{\alpha+1}-A_{i,\alpha,\alpha+1}A_{j,\alpha,\alpha+1}y_{\alpha}y_{\alpha+1}dy_{\alpha+2}$

Now we can skew-symmetrize with respect to $i$ and $j$

(3.117)

A_{i,\alpha+1,\beta}A_{j,\alpha+2,\gamma}y_{\beta}y_{\gamma}dy_{\alpha}\wedge dx_{i}\wedge dx_{j}=\frac{1}{2}(A_{i,\alpha+1,\beta}A_{j,\alpha+2,\gamma}-A_{j,\alpha+1,\beta}A_{i,\alpha+2,\gamma})y_{\beta}y_{\gamma}dy_{\alpha}\wedge dx_{i}\wedge dx_{j}

Correspondingly by skew-symmetrizing each term of (3.116) with respect to $i$ and $j$ , we get

(3.118)		$\displaystyle\frac{1}{2}(A_{i,\alpha+1,\beta}A_{j,\alpha+2,\gamma}-A_{j,\alpha+1,\beta}A_{i,\alpha+2,\gamma})y_{\beta}y_{\gamma}dy_{\alpha}$
	$\displaystyle=$	$\displaystyle\frac{1}{2}(A_{i,\alpha,\alpha+1}A_{j,\alpha,\alpha+2}-A_{i\alpha,\alpha+2}A_{j\alpha,\alpha+1})y_{\alpha}(y_{\alpha}dy_{\alpha}+y_{\alpha+1}dy_{\alpha+1}+y_{\alpha+2}dy_{\alpha+2})$
	$\displaystyle=$	$\displaystyle\frac{1}{2}(A_{i\alpha,\alpha+1}A_{j\alpha,\alpha+2}-A_{i\alpha,\alpha+2}A_{j\alpha,\alpha+1})y_{\alpha}\cdot rdr$

∎

Step 2. In this step will explicitly compute the singular (unbounded) terms of $\Delta\phi_{1}$ . Mainly, we will prove the following proposition.

Proposition 3.15.

Let $\phi_{1}$ be the $3$ -form defined in (3.112), then $\Delta\phi_{1}$ has the following expansion,

	$\displaystyle\Delta\phi_{1}=$	$\displaystyle-\frac{H^{\alpha}y_{\alpha}}{2r^{3}}dy_{1}\wedge dy_{2}\wedge dy_{3}-\Omega_{ij\alpha\beta}(\frac{1}{4r}dy_{\widehat{\alpha\beta}}+\frac{1}{4r^{2}}y_{\widehat{\alpha\beta}}dr)\wedge dx_{i}\wedge dx_{j}$
(3.119)			$\displaystyle+A_{ij\alpha\beta}(\frac{1}{4r^{2}}y_{\widehat{\alpha\beta}}dr-\frac{1}{4r}dy_{\widehat{\alpha\beta}})\wedge dx_{i}\wedge dx_{j}+r^{-5}\Pi_{3}^{(4)}+O^{\prime}(1).$

Proof.

The proof consists of two steps.

The first step focuses on the computation for $d^{*}d\phi_{1}$ . Starting with the expansion of $\phi_{1}$ in (3.113), we have

(3.120)

d\phi_{1}=\frac{-1+H_{\alpha}y_{\alpha}+\widetilde{O}(r^{2})}{2r^{3}}\cdot rdr\wedge\dvol_{N}+\frac{1}{2r}d\Big((1-H^{\alpha}y_{\alpha}+{\widetilde{O}}(r^{2}))\dvol_{N}\Big).

To deal with the first term, we use Lemma 3.11 and (3.13) in Lemma 3.2, then

(3.121)	$\displaystyle y_{\alpha}\eta_{\alpha}$	$\displaystyle=$	$\displaystyle y_{\alpha}dy_{\alpha}-y_{\alpha}p_{i\alpha}dx_{i}$
		$\displaystyle=$	$\displaystyle rdr-y_{\alpha}g_{i\alpha}dx_{i}+\widetilde{O}(r^{4})$
		$\displaystyle=$	$\displaystyle rdr+\widetilde{O}(r^{4}),$

which yields

(3.122)

rdr\wedge\dvol_{N}=(y_{\alpha}\eta_{\alpha})\wedge\dvol_{N}+\widetilde{O}(r^{4})=\widetilde{O}(r^{4}).

So it follows that

(3.123)

d\phi_{1}=\frac{1}{2r}d\Big((1-H^{\alpha}y_{\alpha}+\widetilde{O}(r^{2}))\dvol_{N}\Big)+O^{\prime}(r).

It is easy to see that

(3.124)

d(\widetilde{O}(r^{2})\dvol_{N})=\widetilde{O}(r^{2}).

So we obtain

(3.125)

d\phi_{1}=\frac{1}{2r}d\Big((1-H^{\alpha}y_{\alpha})\dvol_{N}\Big)+O^{\prime}(r).

Next we will compute the expansion for $d(\dvol_{N})$ . By definition,

(3.126)

d(\dvol_{N})=d(\eta_{1}\wedge\eta_{2}\wedge\eta_{3})=d\eta_{\alpha}\wedge\eta_{\widehat{\alpha}}.

By (3.86),

(3.127)		$\displaystyle d\eta_{\alpha}$	$\displaystyle=$	$\displaystyle d(p_{i\alpha})\wedge dx_{i}$
(3.127)			$\displaystyle=$	$\displaystyle A_{i\alpha\beta}dy_{\beta}\wedge dx_{i}+A_{ji\alpha\beta}y_{\beta}dx_{j}\wedge dx_{i}+B_{i\alpha\beta\gamma}y_{\beta}dy_{\gamma}\wedge dx_{i}+\widetilde{O}(r^{2}).$

So we have

(3.128)

d(\dvol_{N})=A_{i\alpha\beta}dy_{\beta}\wedge dx_{i}\wedge\eta_{\widehat{\alpha}}+y_{\beta}(A_{ji\alpha\beta}dx_{j}\wedge dx_{i}+B_{i\alpha\beta\gamma}dy_{\gamma}\wedge dx_{i})\wedge dy_{\widehat{\alpha}}+\widetilde{O}(r^{2}).

Now we need to rearrange the above expansion. Since $A_{i\alpha\beta}$ is skew symmetric in $\alpha$ and $\beta$ , we have for $\alpha\in\{1,2,3\}$ ,

(3.129)

A_{i\alpha\alpha}=0,

so the leading order in the first term vanishes, hence

(3.130)	$\displaystyle A_{i\alpha\beta}dy_{\beta}\wedge dx_{i}\wedge\eta_{\widehat{\alpha}}$	$\displaystyle=$	$\displaystyle A_{i\alpha\beta}A_{j\mu\gamma}y_{\gamma}dy_{\beta}\wedge dx_{i}\wedge dx_{j}\wedge dy_{\widehat{\alpha\mu}}+\widetilde{O}(r^{2})$
		$\displaystyle=$	$\displaystyle A_{i\alpha\beta}A_{j\beta\gamma}y_{\gamma}dy_{\widehat{\alpha}}\wedge dx_{i}\wedge dx_{j}+\widetilde{O}(r^{2})$
		$\displaystyle=$	$\displaystyle\frac{1}{2}(A_{i\alpha\beta}A_{j\beta\gamma}-A_{i\gamma\beta}A_{j\beta\alpha})y_{\gamma}dy_{\widehat{\alpha}}\wedge dx_{i}\wedge dx_{j}+\widetilde{O}(r^{2}).$

Therefore,

(3.131)

d(\dvol_{N})=\Omega_{ij\alpha\beta}\cdot y_{\beta}\cdot dy_{\widehat{\alpha}}\wedge dx_{i}\wedge dx_{j}+B_{i\alpha\beta\alpha}\cdot y_{\beta}\cdot dy_{1}\wedge dy_{2}\wedge dy_{3}\wedge dx_{i}+\widetilde{O}(r^{2}).

By (3.92) we have

(3.132)

d(H^{\alpha}y_{\alpha})\wedge\dvol_{N}=(H^{\alpha}A_{i\alpha\beta}-\partial_{i}(H^{\beta}))\cdot y_{\beta}\cdot dy_{1}\wedge dy_{2}\wedge dy_{3}\wedge dx_{i}+\widetilde{O}(r^{2}).

Now substituting (3.131) and (3.132) into (3.125),

	$\displaystyle d\phi_{1}$	$\displaystyle=\frac{1}{2r}\Omega_{ij\alpha\beta}\cdot y_{\beta}\cdot dy_{\widehat{\alpha}}\wedge dx_{i}\wedge dx_{j}$
(3.133)			$\displaystyle+\frac{1}{2r}\Big(B_{i\alpha\beta\alpha}-(H^{\alpha}A_{i\alpha\beta}-\partial_{i}(H^{\beta})\Big)\cdot y_{\beta}\cdot dy_{1}\wedge dy_{2}\wedge dy_{3}\wedge dx_{i}+O^{\prime}(r).$

Now we need to take $d^{*}$ of this. Notice that the leading order of $d^{*}d\phi_{1}$ can be computed by using the operators in the Euclidean case, so we obtain

(3.134)

\displaystyle d^{*}d\phi_{1}

\displaystyle=

\displaystyle\Omega_{ij\alpha\beta}(\frac{1}{2r}dy_{\widehat{\beta\alpha}}-\frac{1}{2r^{3}}y_{\mu}y_{\beta}dy_{\widehat{\mu\alpha}})dx_{i}\wedge dx_{j}+r^{-3}\Pi_{3}^{(2)}+O^{\prime}(1).

In our next step, we will compute $dd^{*}\phi_{1}$ . First,

(3.135)

*\phi_{1}=\frac{1}{2r}\dvol_{T}.

Notice that $d(\dvol_{T})=0$ , so

(3.136)

d*\phi_{1}=-\frac{1}{2r^{3}}\cdot rdr\wedge\dvol_{T}.

By (3.121), $rdr=y_{\alpha}\eta_{\alpha}+{\widetilde{O}}(r^{4})$ , then

(3.137)

d*\phi_{1}=-\frac{y_{\alpha}}{2r^{3}}{}\eta_{\alpha}\wedge\dvol_{T}+O^{\prime}(r).

Applying Item (2) of Lemma 3.13,

(3.138)

*d*\phi_{1}=-\frac{y_{\alpha}}{2r^{3}}{}(1-H^{\beta}y_{\beta}+\widetilde{O}(r^{2}))\eta_{\widehat{\alpha}}+O^{\prime}(r).

So it follows that

(3.139)

\displaystyle d^{*}\phi_{1}

\displaystyle=

\displaystyle-*d*\phi_{1}=\frac{y_{\alpha}}{2r^{3}}{}(1-H^{\beta}y_{\beta})\eta_{\widehat{\alpha}}{}+\frac{\widetilde{O}(r^{2})}{r^{3}}y_{\alpha}dy_{\widehat{\alpha}}+O^{\prime}(r).

Taking $d$ and applying Lemma 3.2,

	$\displaystyle dd^{*}\phi_{1}=$	$\displaystyle(1-H^{\beta}y_{\beta})\Big(-\frac{3y_{\alpha}}{2r^{5}}{}rdr\wedge\eta_{\widehat{\alpha}}+\frac{1}{2r^{3}}d(y_{\alpha}\eta_{\widehat{\alpha}})\Big){}-\frac{H^{\beta}y_{\alpha}}{2r^{3}}\eta_{\widehat{\alpha}}\wedge dy_{\beta}+r^{-5}\Pi_{3}^{(4)}+O^{\prime}(1)$
	$\displaystyle=$	$\displaystyle(1-H^{\beta}y_{\beta})\Big(-\frac{3y_{\alpha}}{2r^{5}}{}rdr\wedge\eta_{\widehat{\alpha}}+\frac{1}{2r^{3}}d(y_{\alpha}\eta_{\widehat{\alpha}})\Big)-\frac{H^{\alpha}y_{\alpha}}{2r^{3}}dy_{1}\wedge dy_{2}\wedge dy_{3}$
(3.140)		$\displaystyle+$	$\displaystyle r^{-5}\Pi_{3}^{(4)}+O^{\prime}(1).$

Now we simplify this expression. By (3.121),

(3.141)

-\frac{3y_{\alpha}}{2r^{5}}{}rdr\wedge\eta_{\widehat{\alpha}}=-\frac{3y_{\alpha}y_{\beta}}{2r^{5}}{}\eta_{\beta}\wedge\eta_{\widehat{\alpha}}+O^{\prime}(1)=-\frac{3}{2r^{3}}{}\dvol_{N}+{O}^{\prime}(1).

Also

(3.142)	$\displaystyle\frac{1}{2r^{3}}d(y_{\alpha}\eta_{\widehat{\alpha}})$	$\displaystyle=$	$\displaystyle\frac{1}{2r^{3}}dy_{\alpha}\wedge\eta_{\widehat{\alpha}}+\frac{1}{2r^{3}}y_{\alpha}d\eta_{\widehat{\alpha}}$
		$\displaystyle=$	$\displaystyle\frac{1}{2r^{3}}(\eta_{{\alpha}}-p_{i\alpha}dx_{i})\wedge\eta_{\widehat{\alpha}}+\frac{1}{2r^{3}}y_{\alpha}(d{\eta_{\alpha+1}}\wedge{\eta_{\alpha+2}}-{\eta_{\alpha+1}}\wedge d{\eta_{\alpha+2}})$
		$\displaystyle=$	$\displaystyle\frac{3}{2r^{3}}\dvol_{N}-\frac{1}{2r^{3}}(A_{i\alpha\beta}y_{\beta}+\frac{1}{2}B_{i\alpha\beta\gamma}y_{\beta}y_{\gamma})dx_{i}\wedge\eta_{\widehat{\alpha}}$
		$\displaystyle+$	$\displaystyle\frac{1}{2r^{3}}y_{\alpha}(d{\eta_{\alpha+1}}\wedge{\eta_{\alpha+2}}-{\eta_{\alpha+1}}\wedge d{\eta_{\alpha+2}})+{O^{\prime}}(1).$

So it follows that

	$\displaystyle dd^{*}\phi_{1}=$	$\displaystyle-\frac{1}{2r^{3}}\Big(A_{i\alpha\beta}\cdot y_{\beta}\cdot dx_{i}\wedge\eta_{\widehat{\alpha}}-y_{\alpha}(d{\eta_{\alpha+1}}\wedge{\eta_{\alpha+2}}-{\eta_{\alpha+1}}\wedge d{\eta_{\alpha+2}})\Big)$
(3.143)		$\displaystyle-$	$\displaystyle\frac{H^{\alpha}y_{\alpha}}{2r^{3}}dy_{1}\wedge dy_{2}\wedge dy_{3}+r^{-5}\Pi_{3}^{(4)}+O^{\prime}(1).$

Next, we will show a crucial cancellation for the first term of the above $dd^{*}\phi_{1}$ , which gives a further order improvement.

Lemma 3.16 (Cancellation Lemma).

		$\displaystyle A_{i\alpha\beta}\cdot y_{\beta}\cdot dx_{i}\wedge\eta_{\widehat{\alpha}}-y_{\alpha}(d{\eta_{\alpha+1}}\wedge{\eta_{\alpha+2}}-{\eta_{\alpha+1}}\wedge d{\eta_{\alpha+2}})$
(3.144)		$\displaystyle=$	$\displaystyle-A_{ij\alpha\beta}y_{\beta}y_{\mu}dy_{\widehat{\mu\alpha}}\wedge dx_{i}\wedge dx_{j}+\Pi_{3}^{(2)}+{\widetilde{O}}(r^{3}).$

Proof.

Directly applying the definition of $\eta_{\alpha}$ , then we have

		$\displaystyle A_{i\alpha\beta}\cdot y_{\beta}\cdot dx_{i}\wedge\eta_{\widehat{\alpha}}$
(3.145)		$\displaystyle=$	$\displaystyle A_{i\alpha\beta}y_{\beta}dx_{i}\wedge dy_{\widehat{\alpha}}+A_{i\alpha\beta}y_{\beta}y_{\gamma}(A_{j,\alpha+1,\gamma}dy_{\alpha+2}-A_{j,\alpha+2,\gamma}dy_{\alpha+1})\wedge dx_{i}\wedge dx_{j}+\widetilde{O}(r^{3}).$

By (3.127), we get

		$\displaystyle y_{\alpha}(d{\eta_{\alpha+1}}\wedge{\eta_{\alpha+2}}-{\eta_{\alpha+1}}\wedge d{\eta_{\alpha+2}})$
	$\displaystyle=$	$\displaystyle y_{\alpha}(A_{i,\alpha+1,\beta}dy_{\beta}\wedge dx_{i}\wedge dy_{\alpha+2}-A_{i,\alpha+2,\beta}dy_{\beta}\wedge dx_{i}\wedge dy_{\alpha+1})$
	$\displaystyle+$	$\displaystyle y_{\alpha}y_{\gamma}(A_{i,\alpha+1,\beta}A_{j,\alpha+2,\gamma}-A_{i,\alpha+2,\beta}A_{j,\alpha+1,\gamma})dy_{\beta}\wedge dx_{i}\wedge dx_{j}$
	$\displaystyle+$	$\displaystyle y_{\alpha}y_{\beta}(A_{ij,\alpha+1,\beta}dy_{\alpha+2}-A_{ij,\alpha+2,\beta}dy_{\alpha+1})\wedge dx_{i}\wedge dx_{j}$
(3.146)		$\displaystyle+$	$\displaystyle\Pi_{3}^{(2)}+\widetilde{O}(r^{3}).$

Rearranging the subscripts of the first groups of terms in (3.146),

(3.147)		$\displaystyle y_{\alpha}(A_{i,\alpha+1,\beta}dy_{\beta}\wedge dx_{i}\wedge dy_{\alpha+2}-A_{i,\alpha+2,\beta}dy_{\beta}\wedge dx_{i}\wedge dy_{\alpha+1})$
(3.148)	$\displaystyle=$	$\displaystyle y_{\alpha}(A_{i,\alpha+1,\alpha}dy_{\alpha}\wedge dx_{i}\wedge dy_{\alpha+2}-A_{i,\alpha+2,\alpha}dy_{\alpha}\wedge dx_{i}\wedge dy_{\alpha+1})$
(3.149)	$\displaystyle=$	$\displaystyle y_{\alpha+2}A_{i,\alpha,\alpha+2}dy_{\alpha+2}\wedge dx_{i}\wedge dy_{\alpha+1}-y_{\alpha+1}A_{i,\alpha,\alpha+1}dy_{\alpha+1}\wedge dx_{i}\wedge dy_{\alpha+2}$
(3.150)	$\displaystyle=$	$\displaystyle A_{i\alpha\beta}y_{\beta}dx_{i}\wedge dy_{\widehat{\alpha}},$

which matches the first term of (3.145). As in the proof of Lemma 3.14, one can see that the second groups of terms in (3.145) and (3.146) are both equal to

(3.151)

(A_{i\alpha,\alpha+1}A_{j\alpha,\alpha+2}-A_{i\alpha,\alpha+2}A_{j\alpha,\alpha+1})y_{\alpha}\cdot rdr\wedge dx_{i}\wedge dx_{j}.

Next, the third group of terms in (3.146) can be rewritten as follows,

(3.152)		$\displaystyle y_{\alpha}y_{\beta}(A_{ij,\alpha+1,\beta}dy_{\alpha+2}-A_{ij,\alpha+2,\beta}dy_{\alpha+1})\wedge dx_{i}\wedge dx_{j}$
	$\displaystyle=$	$\displaystyle A_{ij\alpha\beta}y_{\beta}(y_{\alpha+2}dy_{\alpha+1}-y_{\alpha+1}dy_{\alpha+2})\wedge dx_{i}\wedge dx_{j}$
	$\displaystyle=$	$\displaystyle A_{ij\alpha\beta}y_{\beta}y_{\mu}dy_{\widehat{\mu\alpha}}\wedge dx_{i}\wedge dx_{j}.$

The conclusion just follows.

∎

Now we return to the expansion of $dd^{*}\phi_{1}$ given by (3.143). Applying Lemma 3.16, finally we obtain

	$\displaystyle dd^{*}\phi_{1}$	$\displaystyle=-\frac{H^{\alpha}y_{\alpha}}{2r^{3}}dy_{1}\wedge dy_{2}\wedge dy_{3}+\frac{1}{2r^{3}}A_{ij\alpha\beta}y_{\beta}y_{\mu}dy_{\widehat{\mu\alpha}}\wedge dx_{i}\wedge dx_{j}$
(3.153)			$\displaystyle+r^{-5}\Pi_{3}^{(4)}+O^{\prime}(1).$

In the last step of the proof, we will further simplify $d^{*}d\phi_{1}$ and $dd^{*}\phi_{1}$ . For this purpose, we need the following lemma.

Lemma 3.17.

(3.154)		$\displaystyle\Omega_{ij\alpha\beta}(\frac{1}{2r}dy_{\widehat{\beta\alpha}}-\frac{1}{2r^{3}}y_{\mu}y_{\beta}dy_{\widehat{\mu\alpha}})$	$\displaystyle=-\Omega_{ij\alpha\beta}(\frac{1}{4r}dy_{\widehat{\alpha\beta}}+\frac{1}{4r^{2}}y_{\widehat{\alpha\beta}}dr),$
(3.155)		$\displaystyle\frac{1}{2r^{3}}A_{ij\alpha\beta}y_{\beta}y_{\mu}dy_{\widehat{\mu\alpha}}$	$\displaystyle=A_{ij\alpha\beta}(\frac{1}{4r^{2}}y_{\widehat{\alpha\beta}}dr-\frac{1}{4r}dy_{\widehat{\alpha\beta}}).$

Proof.

We only prove (3.154) because the other equality follows from the same computations. Using the fact that $\Omega_{ij\alpha\beta}=-\Omega_{ij\beta\alpha}$ , we can write out the left hand side as

		$\displaystyle\Omega_{ij\alpha\beta}(\frac{1}{2r}dy_{\widehat{\beta\alpha}}-\frac{1}{2r^{3}}y_{\mu}y_{\beta}dy_{\widehat{\mu\alpha}})$
	$\displaystyle=$	$\displaystyle\Omega_{ij\alpha,\alpha+1}\Big(-\frac{1}{r}dy_{\alpha+2}-\frac{1}{2r^{3}}(y_{\alpha+2}y_{\alpha+1}dy_{y_{\alpha+1}}-y_{\alpha+1}^{2}dy_{\alpha+2})+\frac{1}{2r^{3}}(y_{\alpha}^{2}dy_{\alpha+2}-y_{\alpha}y_{\alpha+2}dy_{\alpha})\Big)$
	$\displaystyle=$	$\displaystyle\Omega_{ij\alpha,\alpha+1}\Big(-\frac{1}{2r}dy_{\alpha+2}-\frac{1}{2r^{2}}y_{\alpha+2}dr\Big)$
(3.156)		$\displaystyle=$	$\displaystyle-\frac{1}{2}\Omega_{ij\alpha,\beta}\Big(\frac{1}{2r}dy_{\widehat{\alpha\beta}}+\frac{1}{2r^{2}}y_{\widehat{\alpha\beta}}dr\Big).$

∎

Applying the above lemma, now (3.134) and (3.153) can be simplified as follows,

(3.157)		$\displaystyle d^{*}d\phi_{1}$	$\displaystyle=-\Omega_{ij\alpha\beta}(\frac{1}{4r}dy_{\widehat{\alpha\beta}}+\frac{1}{4r^{2}}y_{\widehat{\alpha\beta}}dr)\wedge dx_{i}\wedge dx_{j}+r^{-3}\Pi_{3}^{-2}+O^{\prime}(1),$
	$\displaystyle dd^{*}\phi_{1}$	$\displaystyle=-\frac{H^{\alpha}y_{\alpha}}{2r^{3}}dy_{1}\wedge dy_{2}\wedge dy_{3}+A_{ij\alpha\beta}(\frac{1}{4r^{2}}y_{\widehat{\alpha\beta}}dr-\frac{1}{4r}dy_{\widehat{\alpha\beta}})\wedge dx_{i}\wedge dx_{j}$
(3.158)			$\displaystyle+r^{-5}\Pi_{3}^{(4)}+O^{\prime}(1).$

Therefore,

	$\displaystyle\Delta\phi_{1}=$	$\displaystyle(d^{}d+dd^{})\phi_{1}$
	$\displaystyle=$	$\displaystyle-\frac{H^{\alpha}y_{\alpha}}{2r^{3}}dy_{1}\wedge dy_{2}\wedge dy_{3}-\Omega_{ij\alpha\beta}(\frac{1}{4r}dy_{\widehat{\alpha\beta}}+\frac{1}{4r^{2}}y_{\widehat{\alpha\beta}}dr)\wedge dx_{i}\wedge dx_{j}$
(3.159)			$\displaystyle+A_{ij\alpha\beta}(\frac{1}{4r^{2}}y_{\widehat{\alpha\beta}}dr-\frac{1}{4r}dy_{\widehat{\alpha\beta}})\wedge dx_{i}\wedge dx_{j}+r^{-5}\Pi_{3}^{(4)}+O^{\prime}(1).$

The proof is done.

∎

Step 3. In this step we modify $\phi_{1}$ to kill the unbounded terms on the right hand side of (3.119). We first we recall some elementary computations involving the standard Euclidean Hodge Laplacian.

Lemma 3.18.

Let $\Delta_{0}$ be the standard Hodge Laplacian on the Euclidean space $\mathbb{R}^{3}$ , then the following holds:

(1)

Let $\{y_{1},y_{2},y_{3}\}$ be the Cartesian coordinates of $\mathbb{R}^{3}$ , then

(3.160)

\displaystyle\begin{cases}\Delta_{0}r=-\frac{2}{r},\\ \Delta_{0}(\frac{y_{\alpha}y_{\beta}}{r})=\frac{4y_{\alpha}y_{\beta}}{r^{3}},&\alpha\neq\beta,\\ \Delta_{0}((\frac{y_{\alpha}^{2}}{r}-r))=\frac{4y_{\alpha}^{2}}{r^{3}},\\ \Delta_{0}(\frac{y_{\alpha}}{r})=\frac{2y_{\alpha}}{r^{3}}.\end{cases}

(2)

Denote by $\mathcal{P}_{4}$ the space of all homogeneous degree 4 polynomials on $\mathbb{R}^{3}$ , then the operator

(3.161) $\square:\mathcal{P}_{4}\rightarrow\mathcal{P}_{4};f\mapsto r^{5}\Delta_{0}(r^{-3}f)$

is an isomorphism.

Proof.

The first item is a direct calculation. An convenient way to see this is to use the following two facts

(1)

A homogeneous polynomial degree $k$ polynomial restricts to an eigenfunction of the Hodge-Laplacian $\Delta_{S^{2}}$ on the unit sphere, with eigenvalue $k(k+1)$ .
(2)

Given an eigenfunction $h$ of $\Delta_{S^{2}}$ on the unit sphere with eigenvalue $k$ , for any $l$ , we can extend $h$ to a homogeneous function $h_{l}$ on $\mathbb{R}^{3}\setminus\{0\}$ of degree $l$ , and

(3.162) $\Delta_{0}h_{l}=r^{-2}(k-l(l+1))h_{l}$

For the second item it is possible to write down an explicit inverse to $\Delta_{0}$ . Here we provide a quick abstract proof. First we notice $\square:\mathcal{P}_{4}\to\mathcal{P}_{4}$ is a well-defined linear map. This follows from the standard computations

	$\displaystyle r^{5}\Delta_{0}(r^{-3}f)$	$\displaystyle=$	$\displaystyle r^{5}\Delta_{0}(r^{-3})\cdot f-2r^{5}\nabla(r^{-3})\cdot\nabla f+r^{2}\Delta_{0}f$
		$\displaystyle=$	$\displaystyle-6f-3\nabla(r^{2})\cdot\nabla f+r^{2}\Delta_{0}f.$

Since each term in the above formula is a polynomial in $\mathcal{P}_{4}$ , so $\square f\in\mathcal{P}_{4}$ .

Now to prove $\square$ is an isomorphism it suffices to prove it has a trivial kernel in $\mathcal{P}_{4}$ . Let $u\equiv r^{-3}f$ , then $u=O(r)$ for both $r\to 0$ and $r\to\infty$ . If $\Delta_{0}(u)=0$ , then $u$ is harmonic on $\mathbb{R}^{3}\setminus\{0\}$ . The removable singularity theorem implies that $u$ extends smoothly on $\mathbb{R}^{3}$ . Since $u=O(r)$ as $r\to\infty$ , applying the standard derivative estimate for harmonic functions, we conclude $\nabla^{2}u\equiv 0$ . Therefore, $u$ must be a linear function. Noticing $f\in\mathcal{P}_{4}$ , we conclude $f\equiv 0$ . The proof is done.

∎

Next, we want to find a bounded correction $3$ -form $\mathfrak{B}_{0}=O^{\prime}(1)$ such that $\Delta\phi_{1}$ is corrected to a bounded term on $\mathcal{U}\setminus P$ , i.e.,

(3.163)

\Delta(\phi_{1}+\mathfrak{B}_{0})=O^{\prime}(1)\ \text{on}\ \mathcal{U}\setminus P.

Now the main part is to eliminate the unbounded terms in $\Delta\phi_{1}$ which relies on the following explicit calculations for $\Delta\mathfrak{B}_{0}$ . In fact, the leading terms of $\Delta\mathfrak{B}_{0}$ are exactly given by the Euclidean Laplacian $\Delta_{0}$ acting on the normal components such that the explicit computations in Lemma 3.18 can be effectively used in our context. Precisely, we have the following lemma.

Lemma 3.19.

Let $\Delta_{0}$ be the Hodge Laplacian on $\mathbb{R}^{3}$ , then the following holds:

(1)

Denote by $\nu_{y}$ one of the following differential forms $dy_{\alpha}$ , $dy_{\alpha}\wedge dy_{\beta}$ or $dy_{1}\wedge dy_{2}\wedge dy_{3}$ . Similarly, let $\tau_{x}$ be a tangential $p$ -form given by $\tau_{x}\equiv dx_{1}\wedge\ldots\wedge dx_{\alpha_{p}}$ with $0\leq p\leq m-3$ . Let

(3.164) $\omega\equiv f(x)h(y)\nu_{y}\wedge\tau_{x},$

where $f(x)$ is a smooth function defined on $U$ and $h(y)=O^{\prime}(|y|^{k})$ for some $k\in\mathbb{Z}_{+}$ , then

(3.165) $\Delta\omega-f(x)\cdot\Delta_{0}(h(y))\cdot\nu_{y}\wedge\tau_{x}=O^{\prime}(|y|^{k-1}).$
(2)

Let $f$ be a smooth function defined on $U\subset P$ and let

(3.166) $\omega\equiv f(x)\cdot\frac{y_{\alpha}}{r}dy_{1}\wedge dy_{2}\wedge dy_{3},$

then

(3.167) $\displaystyle\Delta\omega=2r^{-2}\omega+r^{-5}\Gamma_{3}^{(4)}+O^{\prime}(1),$

where the definition of the $3$ -form $\Gamma_{3}^{(4)}$ is in (3.45) of Notation 3.9.

Proof.

First, we prove Item (1). By definition, $\Delta=dd^{*}+d^{*}d$ . We only prove the case $\nu_{y}=dy_{\alpha}$ for $1\leq\alpha\leq 3$ and $1\leq p\leq m-3$ . The proof of the remaining cases is identical.

First, we compute $d^{*}d\omega$ .

(3.168)

\displaystyle d\omega

\displaystyle=f(x)\cdot\frac{\partial h(y)}{\partial y_{\beta}}dy_{\beta}\wedge dy_{\alpha}\wedge\tau_{x}+\frac{\partial f(x)}{\partial x_{j}}\cdot h(y)\cdot dx_{j}\wedge\nu_{y}\wedge\tau_{x},

which implies that

	$\displaystyle*d\omega$	$\displaystyle=(-1)^{p}f(x)\cdot\frac{\partial h(y)}{\partial y_{\beta}}dy_{\widehat{\beta\alpha}}\wedge*_{T}(\tau_{x})+O^{\prime}(\|y\|^{k})$
(3.169)			$\displaystyle=(-1)^{p}f(x)\Big(\frac{\partial h(y)}{\partial y_{\alpha-1}}dy_{\alpha+1}-\frac{\partial h(y)}{\partial y_{\alpha+1}}dy_{\alpha-1}\Big)\wedge*_{T}(\tau_{x})+O^{\prime}(\|y\|^{k}).$

Differentiating the above equality,

	$\displaystyle d*d\omega$	$\displaystyle=(-1)^{p}f\cdot\Big(\frac{\partial^{2}h}{\partial y_{\alpha}\partial y_{\alpha-1}}dy_{\alpha}\wedge dy_{\alpha+1}-\frac{\partial^{2}h}{\partial y_{\alpha}\partial y_{\alpha+1}}dy_{\alpha}\wedge dy_{\alpha-1}$
(3.170)			$\displaystyle+\Big(\frac{\partial^{2}h}{\partial y_{\alpha-1}^{2}}+\frac{\partial^{2}h}{\partial y_{\alpha+1}^{2}}\Big)dy_{\alpha-1}\wedge dy_{\alpha+1}\Big)\wedge*_{T}(\tau_{x})+O^{\prime}(\|y\|^{k-1}).$

Then it follows that

	$\displaystyle d^{*}d\omega$	$\displaystyle=(-1)^{mp+m+1}dd\omega$
		$\displaystyle=f\cdot\Big(\frac{\partial^{2}h}{\partial y_{\alpha}\partial y_{\alpha-1}}dy_{\alpha-1}+\frac{\partial^{2}h}{\partial y_{\alpha}\partial y_{\alpha+1}}dy_{\alpha+1}-\Big(\frac{\partial^{2}h}{\partial y_{\alpha-1}^{2}}+\frac{\partial^{2}h}{\partial y_{\alpha+1}^{2}}\Big)dy_{\alpha}\Big)\wedge\tau_{x}$
(3.171)			$\displaystyle+O^{\prime}(\|y\|^{k-1}).$

On the other hand,

(3.172)

\displaystyle*\omega=f(x)h(y)dy_{\widehat{\alpha}}\wedge*_{T}(\tau_{x}),

which implies

(3.173)

d*\omega=f(x)\frac{\partial h}{\partial y_{\alpha}}\dvol_{N}\wedge*_{T}(\tau_{x})+O^{\prime}(|y|^{k}).

So it follows that

(3.174)

d^{*}\omega=(-1)^{mp+1}*d*\omega=-f\cdot\frac{\partial h}{\partial y_{\alpha}}\cdot\tau_{x}+O^{\prime}(|y|^{k}),

and hence

(3.175)

dd^{*}\omega=-f\cdot\Big(\frac{\partial^{2}h}{\partial y_{\alpha-1}\partial y_{\alpha}}dy_{\alpha-1}+\frac{\partial^{2}h}{\partial y_{\alpha}^{2}}dy_{\alpha}+\frac{\partial^{2}h}{\partial y_{\alpha+1}\partial y_{\alpha}}dy_{\alpha+1}\Big)\cdot dy_{\alpha}\wedge\tau_{x}+O^{\prime}(|y|^{k-1}).

Therefore, combining (3.171) and (3.175),

(3.176)

\Delta\omega=(d^{*}d+dd^{*})\omega=-f\cdot(\Delta_{0}h(y))dy_{\alpha}\wedge\tau_{x}+O^{\prime}(|y|^{k-1}),

where $\Delta_{0}(h(y))=-\frac{\partial^{2}h}{\partial y_{1}^{2}}-\frac{\partial^{2}h}{\partial y_{2}^{2}}-\frac{\partial^{2}h}{\partial y_{3}^{2}}$ . The proof of (1) is done.

Now we prove Item (2). Let $\omega=f\cdot\frac{y_{\alpha}}{r}dy_{1}\wedge dy_{2}\wedge dy_{3}$ and the first step is to compute the term $d^{*}d\omega$ . By Lemma 3.2, $dr=\frac{y_{\gamma}dy_{\gamma}}{r}$ , then

(3.177)

d(\frac{y_{\alpha}}{r})\wedge dy_{1}\wedge dy_{2}\wedge dy_{3}=0.

This implies that

(3.178)

d\omega=\frac{\partial f}{\partial x_{i}}\cdot\frac{y_{\alpha}}{r}dx_{i}\wedge dy_{1}\wedge dy_{2}\wedge dy_{3},

and hence

(3.179)

*d\omega=-\frac{\partial f}{\partial x_{i}}\cdot\frac{y_{\alpha}}{r}*_{T}(dx_{i})+O^{\prime}(r).

Differentiating the above equality and applying Lemma 3.2 again,

(3.180)

\displaystyle d*d\omega=-\frac{\partial f}{\partial x_{i}}\Big(\frac{dy_{\alpha}}{r}-\frac{y_{\alpha}y_{\beta}\cdot dy_{\beta}}{r^{3}}\Big)*(dx_{i})+O^{\prime}(1).

It follows that

(3.181)

\displaystyle*d*d\omega=(-1)^{m+1}\cdot\frac{\partial f}{\partial x_{i}}\cdot\frac{dy_{\widehat{\alpha}}\wedge dx_{i}}{r}+(-1)^{m}\frac{\partial f}{\partial x_{i}}\cdot\frac{y_{\alpha}y_{\beta}}{r^{3}}\cdot dy_{\widehat{\beta}}\wedge dx_{i}+O^{\prime}(1)

Therefore,

	$\displaystyle d^{*}d\omega$	$\displaystyle=(-1)^{m+1}dd\omega$
(3.182)			$\displaystyle=\frac{\partial f}{\partial x_{i}}\cdot\frac{dy_{\widehat{\alpha}}\wedge dx_{i}}{r}-\frac{\partial f}{\partial x_{i}}\cdot\frac{y_{\alpha}y_{\beta}}{r^{3}}\cdot dy_{\widehat{\beta}}\wedge dx_{i}+O^{\prime}(1).$

Now we compute $dd^{*}\omega$ . By Lemma 3.13 and the expansion of $\dvol_{N}$ in (3.92),

	$\displaystyle*(dy_{1}\wedge dy_{2}\wedge dy_{3})$	$\displaystyle=*(\dvol_{N}+A_{i\gamma\beta}y_{\gamma}dy_{\widehat{\gamma}}\wedge dx_{i})+\widetilde{O}(r^{2})$
(3.183)			$\displaystyle=(1+H^{\beta}y_{\beta})\dvol_{T}+A_{i\gamma\beta}y_{\gamma}dy_{\gamma}\wedge*_{T}(dx_{i})+\widetilde{O}(r^{2}),$

so we have

	$\displaystyle*\omega$	$\displaystyle=f\cdot\frac{y_{\alpha}}{r}\cdot\dvol_{T}+f\cdot H^{\beta}\cdot\frac{y_{\alpha}y_{\beta}}{r}\cdot\dvol_{T}+f\cdot A_{i\gamma\beta}\frac{y_{\alpha}y_{\gamma}}{r}dy_{\gamma}\wedge*_{T}(dx_{i})+O^{\prime}(r^{2})$
(3.184)			$\displaystyle\equiv\FT_{1}+\FT_{2}+\FT_{3}+O^{\prime}(r^{2}).$

By collecting the leading terms, it is easy to compute the leading term in the above equality,

(3.185)		$\displaystyle d*d(\FT_{1})$	$\displaystyle=\frac{\partial f}{\partial x_{i}}\Big(\frac{dx_{i}\wedge dy_{\widehat{\alpha}}}{r}-\frac{y_{\alpha}y_{\beta}dx_{i}\wedge dy_{\widehat{\beta}}}{r^{3}}\Big)-2\omega+O^{\prime}(1)$
(3.186)		$\displaystyle d*d(\FT_{2})$	$\displaystyle=r^{-5}\Pi_{3}^{(4)},\ d*d(\FT_{3})=r^{-5}\Pi_{3}^{(4)}.$

Therefore,

(3.187)

dd^{*}\omega=-\frac{\partial f}{\partial x_{i}}\cdot\frac{dx_{i}\wedge dy_{\widehat{\alpha}}}{r}+\frac{\partial f}{\partial x_{i}}\cdot\frac{y_{\alpha}y_{\beta}}{r^{3}}\cdot dx_{i}\wedge dy_{\widehat{\beta}}+2\omega+r^{-5}\Pi_{3}^{(4)}+O^{\prime}(1).

By (3.182) and (3.187) we obtain the expansion

(3.188)

\displaystyle\Delta\omega=(d^{*}d+dd^{*})\omega=2\omega+r^{-5}\Pi_{3}^{(4)}+O^{\prime}(1).

So the proof is done.

∎

Now we finish Step 2 by proving the following

Proposition 3.20.

There is some $3$ -form $\Lambda_{3}^{(4)}$ (given in Notation 3.9) such that if we choose

	$\displaystyle\mathfrak{B}_{0}$	$\displaystyle\equiv\frac{H^{\alpha}y_{\alpha}}{4r}dy_{1}\wedge dy_{2}\wedge dy_{3}$
(3.189)			$\displaystyle+\Big(\Omega_{ij\alpha\beta}(\frac{1}{16}y_{\widehat{\alpha\beta}}dr-\frac{3}{16}rdy_{\widehat{\alpha\beta}})-\frac{1}{16}A_{ij\alpha\beta}(y_{\widehat{\alpha\beta}}dr+rdy_{\widehat{\alpha\beta}})\Big)\wedge dx_{i}\wedge dx_{j}+r^{-3}\Lambda_{3}^{(4)},$

then the corrected $3$ -form of $\phi_{1}$ ,

(3.190)

\displaystyle\phi_{2}\equiv

\displaystyle\phi_{1}+\mathfrak{B}_{0}

satisfies

(3.191)

\Delta\phi_{2}=O^{\prime}(1)\ \text{on}\ \mathcal{U}\setminus P,

and has the expansion

	$\displaystyle\phi_{2}=$	$\displaystyle\frac{1}{2r}(1-\frac{H^{\alpha}y_{\alpha}}{2})dy_{1}\wedge dy_{2}\wedge dy_{3}+\frac{1}{2r}y_{\beta}A_{i\alpha\beta}dx_{i}\wedge dy_{\widehat{\alpha}}-\frac{1}{4}A_{ij\alpha\beta}r\cdot dy_{\widehat{\alpha\beta}}\wedge dx_{i}\wedge dx_{j}$
(3.192)			$\displaystyle+\frac{3}{16}(A_{i\alpha,\alpha+1}A_{j\alpha,\alpha+2}-A_{i\alpha,\alpha+2}A_{j\alpha,\alpha+1})d(ry_{\alpha})\wedge dx_{i}\wedge dx_{j}+r^{-3}\Pi_{3}^{(4)}+O^{\prime}(r^{2}).$

Proof.

Let $\mathfrak{b}_{0}\equiv\frac{H^{\alpha}y_{\alpha}}{4r}dy_{1}\wedge dy_{2}\wedge dy_{3}$ , then Item (2) of Lemma 3.19 tells us that

(3.193)

\Delta\mathfrak{b}_{0}=\frac{H^{\alpha}y_{\alpha}}{2r^{3}}dy_{1}\wedge dy_{2}\wedge dy_{3}+r^{-5}\Gamma_{3}^{(4)}+O^{\prime}(1).

Let $\Pi^{(4)}$ be the $3$ -form in the expansion of $\Delta\phi_{1}$ given by (3.119) in Proposition 3.15.

Next, Lemma 3.18 and Lemma 3.19 tell us that there are $3$ -forms $\widehat{\Gamma}_{3}^{(4)}$ and $\widehat{\Pi}_{3}^{(4)}$ which are also of the form as in (3.45) such that

(3.194)		$\displaystyle\Delta(r^{-3}\widehat{\Gamma}_{3}^{(4)})=-r^{-5}\Gamma_{3}^{(4)}+O^{\prime}(1),$
(3.195)		$\displaystyle\Delta(r^{-3}\widehat{\Pi}_{3}^{(4)})=-r^{-5}\Pi_{3}^{(4)}+O^{\prime}(1).$

Now let

(3.196)

\mathfrak{b}_{1}\equiv r^{-3}\widehat{\Gamma}_{3}^{(4)}+r^{-3}\widehat{\Pi}_{3}^{(4)},

then the correction term $\mathfrak{b}_{0}+\mathfrak{b}_{1}$ is chosen as the above such that $\Delta(\mathfrak{b}_{0}+\mathfrak{b}_{1})$ in fact eliminates the $O^{\prime}(r^{-2})$ -term and implicit $O^{\prime}(r^{-1})$ -terms in the expansion of $\Delta\phi_{1}$ (see Proposition 3.15).

In the following, we will make a further correction such that those explicit $O^{\prime}(r^{-1})$ -terms will be cancelled out as well. In fact, we define

(3.197)

\mathfrak{b}_{2}\equiv\Big(\Omega_{ij\alpha\beta}(\frac{1}{16}y_{\widehat{\alpha\beta}}dr-\frac{3}{16}rdy_{\widehat{\alpha\beta}})-\frac{1}{16}A_{ij\alpha\beta}(y_{\widehat{\alpha\beta}}dr+rdy_{\widehat{\alpha\beta}})\Big)\wedge dx_{i}\wedge dx_{j},

applying Lemma 3.18 and Lemma 3.19 again, then

(3.198)

\Delta\mathfrak{b}_{2}=\Omega_{ij\alpha\beta}\Big(\frac{1}{4r}dy_{\widehat{\beta\alpha}}+\frac{y_{\widehat{\alpha\beta}}}{4r^{2}}dr\Big)\wedge dx_{i}\wedge dx_{j}-A_{ij\alpha\beta}\Big(\frac{y_{\widehat{\alpha\beta}}}{4r^{2}}dr-\frac{1}{4r}dy_{\widehat{\alpha\beta}}\Big)\wedge dx_{i}\wedge dx_{j},

and hence

(3.199)

\Delta(\phi_{1}+\mathfrak{b}_{0}+\mathfrak{b}_{1}+\mathfrak{b}_{2})=O^{\prime}(1).

Therefore, it suffices to choose the correction term

(3.200)

\mathfrak{B}_{0}\equiv\mathfrak{b}_{0}+\mathfrak{b}_{1}+\mathfrak{b}_{2},

which gives $\Delta(\phi_{1}+\mathfrak{B}_{0})=O^{\prime}(1)$ .

Notice that, $\mathfrak{b}_{2}$ has a further cancellation,

	$\displaystyle\mathfrak{b}_{2}=$	$\displaystyle\Big(\Omega_{ij\alpha\beta}(\frac{1}{16}y_{\widehat{\alpha\beta}}dr-\frac{3}{16}rdy_{\widehat{\alpha\beta}})-\frac{1}{16}A_{ij\alpha\beta}(y_{\widehat{\alpha\beta}}dr+rdy_{\widehat{\alpha\beta}})\Big)\wedge dx_{i}\wedge dx_{j},$
	$\displaystyle=$	$\displaystyle-\frac{1}{4}A_{ij\alpha\beta}rdy_{\widehat{\alpha\beta}}\wedge dx_{i}\wedge dx_{j}-\frac{1}{16}(A_{i,\alpha,\alpha+1}A_{j,\alpha,\alpha+2}-A_{i,\alpha,\alpha+2}A_{j,\alpha,\alpha+1})y_{\alpha}dr\wedge dx_{i}\wedge dx_{j}$
(3.201)			$\displaystyle+\frac{3}{16}(A_{i,\alpha,\alpha+1}A_{j,\alpha,\alpha+2}-A_{i,\alpha,\alpha+2}A_{j,\alpha,\alpha+1})rdy_{\alpha}\wedge dx_{i}\wedge dx_{j}.$

Therefore,

	$\displaystyle\phi_{2}=$	$\displaystyle\phi_{1}+\mathfrak{B}_{0}$
	$\displaystyle=$	$\displaystyle\frac{1}{2r}(1-\frac{H^{\alpha}y_{\alpha}}{2})dy_{1}\wedge dy_{2}\wedge dy_{3}+\frac{1}{2r}y_{\beta}A_{i\alpha\beta}dx_{i}\wedge dy_{\widehat{\alpha}}-\frac{1}{4}A_{ij\alpha\beta}r\cdot dy_{\widehat{\alpha\beta}}\wedge dx_{i}\wedge dx_{j}$
(3.202)			$\displaystyle+\frac{3}{16}(A_{i\alpha,\alpha+1}A_{j\alpha,\alpha+2}-A_{i\alpha,\alpha+2}A_{j\alpha,\alpha+1})d(ry_{\alpha})\wedge dx_{i}\wedge dx_{j}+r^{-3}\Pi_{3}^{(4)}+O^{\prime}(r^{2}),$

and

(3.203)

\Delta\phi_{2}=O^{\prime}(1).

∎

Step 4. In this step we compute $\Delta\phi_{2}$ as a current on $\mathcal{U}$ .

Lemma 3.21.

In $\mathcal{U}$ , we have

(3.204)

\Delta\phi_{2}=2\pi\delta_{U}+O^{\prime}(1).

Proof.

Suppose we are given a compactly supported test form $\chi\in\Omega_{0}^{m-3}(\mathcal{U})$ , then we apply integration by parts once and we have

(3.205)

(\phi_{2},\Delta\chi)=\int_{\mathcal{U}}\phi_{2}\wedge(dd^{*}+d^{*}d)\chi=\int_{\mathcal{U}}d\phi_{2}\wedge d^{*}\chi-\int_{\mathcal{U}}d^{*}\phi_{2}\wedge d\chi.

Here there is no boundary term because $\phi_{2}=O(r^{-1})$ . Notice that (3.133) and (3.190) implies $d\phi_{2}=O^{\prime}(1)$ , so

(3.206)

\int_{\mathcal{U}}d\phi_{2}\wedge d^{*}\chi=\int_{\mathcal{U}}d^{*}d\phi_{2}\wedge\chi.

On the other hand, by (3.139),

(3.207)

d^{*}\phi_{2}=-\frac{y_{\alpha}}{2r^{3}}{}dy_{\hat{\alpha}}+\zeta,

where $\zeta$ is a $2$ -form satisfying $\zeta=O^{\prime}(r^{-1})$ . Denote by $S_{\epsilon}^{2}$ the normal geodesic sphere bundle $\{r=\epsilon\}$ , then we get that

(3.208)		$\displaystyle-\int_{\mathcal{U}}d^{*}\phi_{2}\wedge d\chi$	$\displaystyle=$	$\displaystyle\int_{\mathcal{U}}dd^{}\phi_{2}\wedge\chi+\int_{S_{\epsilon}^{2}}d^{}\phi_{2}\wedge\chi$
(3.208)			$\displaystyle=$	$\displaystyle\int_{\mathcal{U}}dd^{*}\phi_{2}\wedge\chi+\lim_{\epsilon\rightarrow 0}\frac{1}{2\epsilon^{3}}{}\int_{S_{\epsilon}}(y_{\alpha}dy_{\hat{\alpha}}+\epsilon^{2}\zeta)\wedge\chi.$

By direct calculation of the last term on the right hand side we obtain

(3.209)

-\int_{\mathcal{U}}d^{*}\phi_{2}\wedge d\chi=\int_{\mathcal{U}}dd^{*}\phi_{2}\wedge\chi+2\pi{}\int_{P}\chi.

This concludes the proof. ∎

Step 5. Now we solve the Laplace equation with right hand side in $O^{\prime}(1).$

Lemma 3.22.

Given a local 3-form $v$ defined on a neighborhood $\mathcal{U}$ of $p$ in $Q$ with $v=O^{\prime}(1)$ , then there is some smaller neighborhood $\mathcal{V}\subset\subset\mathcal{U}$ such that there exists a local solution $T$ to the equation

(3.210)

\Delta T=v,

with $T=O^{\prime}(r^{2})$ . Here $r$ is the distance to the submanifold $P$ .

Proof.

We just need to establish the following:

(1)

(General derivatives estimate) For each $k\in\mathbb{N}$ and $\epsilon>0$ , it holds that

(3.211) $|\nabla^{k+2}T|=O(r^{-(k+\epsilon)}).$
(2)

(Mixed derivatives estimate) For each $k\in\mathbb{N}$ , $\ell\in\mathbb{N}$ and $\epsilon>0$ , it holds that

(3.212) $|(\nabla^{t})^{k}\nabla^{\ell+2}T|=O(r^{-(\ell+\epsilon)}),$

where $\nabla^{t}$ denotes the tangential derivative.

The above estimates will be proved by induction.

First, we will prove the following order estimate for $\nabla^{2}T$ in a smaller neighborhood $\mathcal{U}^{\prime}\subset\subset\mathcal{U}$

(3.213)

|\nabla^{2}T|=O(r^{-\epsilon}).

This can be viewed as the base step for carrying out the inductive argument.

To begin with, by definition, for any $p>0$ , we have $|v|_{L^{p}(\mathcal{U})}\leq C_{p}$ . Applying the standard elliptic $W^{2,p}$ -estimate, for each $p>0$ , there is some constant $C_{p}>0$ such that in a smaller neighborhood $\mathcal{U}_{1}\subset\subset\mathcal{U}$ such that

(3.214)

\|\nabla^{2}T\|_{L^{p}(\mathcal{U}_{1})}\leq C_{p}.

Then Sobolev embedding theorem tells us that

(3.215)

T\in W^{2,p}(\mathcal{U}_{1})\cap C^{1,\alpha}(\mathcal{U}_{1})

for any $p>1$ and $0<\alpha<1$ .

To prove (3.213), we need to differentiate the equation, which schematically yields that

(3.216)

\Delta\nabla^{2}T+Q_{1}*\nabla^{3}T+Q_{2}*\nabla^{2}T+Q_{3}*\nabla T=w,

where $|w|=O^{\prime}(r^{-2})$ and $Q_{i}$ ’s are smooth terms arising from differentiating the coefficients of $\Delta$ . The above equation can be viewed as an elliptic system in terms of the Hessian of $T$ . Let $\Phi\equiv\nabla^{2}T$ , noticing $\nabla T\in C^{\alpha}(\mathcal{U}_{1})$ , so the terms involving $\nabla T$ can be absorbed to the right hand side of the equation. Then $\Phi$ can be treated as vector valued functions, once we fix a local frame. So it follows that

(3.217)

\Delta\Phi+Q_{1}*\nabla\Phi+Q_{2}*\Phi=\eta,

where $|\eta|=O^{\prime}(r^{-2})$ . Since $\eta$ has unbounded $L^{p}$ -norm for large $p$ , the standard $W^{2,p}$ -estimate for $\Phi$ does not directly apply.

For improving the regularity of $\nabla^{2}T$ , we will rescale the metric $g$ . For each $x$ in an even smaller neighborhood $\mathcal{U}_{2}$ with $r(x)=r_{x}$ , we rescale the metric $g$ in $B_{r_{x}}(x)$ by letting

(3.218)

\tilde{g}=(r_{x})^{-2}g,

then the following equation holds in the rescaled geodesic ball $B_{1}^{\tilde{g}}(x)$ ,

(3.219)

\widetilde{\Delta}\widetilde{\Phi}+r_{x}\widetilde{Q}_{1}*\widetilde{\nabla}\widetilde{\Phi}+(r_{x})^{2}\widetilde{Q}_{2}*\widetilde{\Phi}=\tilde{\eta},\quad|\tilde{\eta}|=O^{\prime}(1),

where $\widetilde{\Phi}(y)=\Phi(r_{x}\cdot y)$ for each $y\in B_{1}^{\tilde{g}}(x)$ . In the above equation, all the coefficients are uniformly bounded independent of $x$ . Since we have shown $|\Phi|_{L^{p}}\leq C_{p}$ in (3.214), so simple rescaling gives rise to the following estimate for any $p>0$ ,

(3.220)

|\widetilde{\Phi}|_{L^{p}(B_{1}^{\tilde{g}}(x))}\leq C_{p}\cdot(r_{x})^{-\frac{n}{p}}.

Now applying the $W^{2,p}$ -estimate for $\widetilde{\Phi}$ , then for each $p>0$

(3.221)

|\widetilde{\Phi}|_{W^{2,p}(B_{1/2}^{\tilde{g}}(x))}\leq C_{p}\cdot(r_{x})^{-\frac{n}{p}}

with $C_{p}>0$ independent of $x$ . By the Sobolev embedding

(3.222)

|\widetilde{\Phi}|_{C^{1,\alpha}(B_{1/4}^{\tilde{g}}(x))}\leq C_{\alpha,p}\cdot(r_{x})^{-\frac{n}{p}}

with $C_{\alpha,p}>0$ independent of $x$ . Scale back to the original metric, for any $p>0$ , there is some $C_{\alpha,p}>0$ independent of the base point $x\in\mathcal{U}_{2}$ such that

(3.223)

|\Phi|_{L^{\infty}(B_{r_{x}/4}(x))}+|r_{x}\nabla\Phi|_{L^{\infty}(B_{r_{x}/4}(x))}\leq C_{\alpha,p}\cdot(r_{x})^{-\frac{n}{p}}.

This completes the proof of (3.213).

Now we will finish the proof of Item (1) by using the induction. Based on (3.223), the key induction step is to prove the following: Given any $\ell\in\mathbb{Z}_{+}$ , if for each $\epsilon>0$ and $0\leq k\leq\ell-1$ ,

(3.224)

|\nabla^{k}\Phi|=O(r^{-(k+\epsilon)}),\quad|\widetilde{\nabla}^{k}\widetilde{\Phi}|_{W^{2,p}(B_{1/2}^{\tilde{g}}(x))}\leq C_{k,p,\epsilon}\cdot(r_{x})^{-\epsilon},

then for each $\epsilon>0$ , we have

(3.225)

|\nabla^{\ell}\Phi|=O(r^{-(\ell+\epsilon)}),\quad|\widetilde{\nabla}^{\ell}\widetilde{\Phi}|_{W^{2,p}(B_{1/4}^{\tilde{g}}(x))}\leq C_{\ell,p,\epsilon}\cdot(r_{x})^{-\epsilon}.

Indeed, then differentiating (3.216) by $\nabla^{k}$ ,

(3.226)

\Delta(\nabla^{\ell}\Phi)+\sum\limits_{j=1}^{\ell+1}Q_{j}*\nabla^{j}\Phi=w_{\ell},

where $|w_{\ell}|=O^{\prime}(r^{-(\ell+2)})$ . As before, we rescale the metric $g$ by taking $\tilde{g}=(r_{x})^{-2}g$ , then

(3.227)

\widetilde{\Delta}(\widetilde{\nabla}^{\ell}\widetilde{\Phi})+\sum\limits_{j=1}^{\ell+1}(r_{x})^{\ell-j+2}\cdot Q_{j}*\nabla^{j}\widetilde{\Phi}=\tilde{w}_{\ell},

where $|\tilde{w}_{\ell}|=O^{\prime}(1)$ . Let $k=\ell-1$ , applying the induction hypothesis (3.224) and Sobolev embedding, we have

(3.228)

|\widetilde{\nabla}^{\ell}\widetilde{\Phi}|_{L^{\infty}(B_{1/2}^{\tilde{g}}(x))}=O((r_{x})^{-\epsilon}).

The above enables us to apply the $W^{2,p}$ -elliptic estimate, so we obtain the following estimate for each $\epsilon>0$ ,

(3.229)

|\widetilde{\nabla}^{\ell}\widetilde{\Phi}|_{W^{2,p}(B_{1/4}^{\tilde{g}}(x))}\leq C_{\ell,p,\epsilon}\cdot(r_{x})^{-\epsilon},

where $C_{\ell,p,\epsilon}>0$ is independent of the base point $x$ . Applying the Sobolev embedding $W^{2,p}\subset C^{1,\alpha}$ and scaling back to the original metric $g$ ,

(3.230)

|\nabla^{\ell}\Phi|_{L^{\infty}(B_{r_{x}/8}(x))}\leq C_{\ell,\epsilon}\cdot(r_{x})^{-(\ell+\epsilon)}

for each $\epsilon>0$ . So we complete the proof of Item (1).

Now we are ready to finish the proof of Item (2). We only focus on the case $\ell=0$ and the case for $\ell>0$ can be directly achieved by applying the above rescaling arguments. To this end, we need the following claim for the tangential derivatives estimate.

Claim. Let $|\widetilde{\Phi}|_{W^{2,p}(B_{1/2}^{\tilde{g}}(x))}\leq C_{p,\epsilon}\cdot(r_{x})^{-\epsilon}$ for any $\epsilon>0$ and for any $x\in\mathcal{U}$ . Assume that $\widetilde{\Phi}$ solves the elliptic equation

(3.231)

\Delta\widetilde{\Phi}+Q_{1}*\nabla\widetilde{\Phi}+Q_{2}*\widetilde{\Phi}=\zeta,

where $Q_{i}$ ’s are smooth coefficients, $|\zeta|\in O^{\prime}(1)$ . Then for any $x\in\mathcal{U}$ , the estimate

(3.232)

|(\nabla^{t})^{k}\widetilde{\Phi}|_{W^{2,p}(B_{1/8}^{\tilde{g}}(x))}\leq C_{k,p,\epsilon}\cdot(r_{x})^{-\epsilon},

holds for all $k\geq 1$ , $p>1$ and $\epsilon>0$ .

Taking the first tangential derivative $\nabla^{t}$ for $\Delta\widetilde{\Phi}$ ,

(3.233)

(\nabla^{t})(\Delta\widetilde{\Phi})=\Delta\nabla^{t}\widetilde{\Phi}+Q_{1}*\nabla^{2}\widetilde{\Phi}+Q_{2}*\nabla\widetilde{\Phi},

where $Q_{i}$ ’s are smooth functions. Hence differentiating (3.231) once by the tangential derivative $\nabla^{t}$ , we have

(3.234)

\Delta(\nabla^{t}\widetilde{\Phi})+Q_{1}*\nabla^{2}\widetilde{\Phi}+Q_{2}*\nabla\widetilde{\Phi}=w_{1},

where $Q_{i}$ ’s are smooth functions, $w_{1}\equiv\nabla^{t}\zeta$ and $|w_{1}|\in O^{\prime}(1)$ . Since we have already assumed $|\nabla^{2}\widetilde{\Phi}|_{L^{p}(B_{1/2}^{\tilde{g}}(x))}\leq C_{p,\epsilon}\cdot(r_{x})^{-\epsilon}$ for all $\epsilon>0$ , applying the standard $W^{2,p}$ -estimate, then for any $p>1$ and $\epsilon>0$ ,

(3.235)

|\nabla^{t}\widetilde{\Phi}|_{W^{2,p}(B_{1/3}^{\tilde{g}}(x))}\leq C_{p,\epsilon}\cdot(r_{x})^{-\epsilon}.

Now we prove the higher order mixed derivatives estimate by induction. Repeat taking the tangential derivatives and let $\Psi^{(k)}\equiv(\nabla^{t})^{k}\widetilde{\Phi}$ for all $k>1$ . Assume that $|\Psi^{(j)}|_{W^{2,p}(B_{1/4}^{\tilde{g}}(x))}\leq C_{k,p,\epsilon}(r_{x})^{-\epsilon}$ holds for all $j\leq k-1$ and $p>1$ , then

(3.236)

\Delta(\Psi^{(k)})+\sum_{\begin{subarray}{c}\mu=1,2\\ 1\leq\mu+\nu\leq k\end{subarray}}Q_{\mu\nu}*\nabla^{\mu}(\Psi^{(\nu)})=w_{k},

where $w_{k}\equiv(\nabla^{t})^{k}\zeta$ and $|w_{k}|=O^{\prime}(1)$ . Applying the induction hypothesis, it follows that for any $k\geq 1$ ,

(3.237)

|\Delta(\Psi^{(k)})|_{L^{p}(B_{1/4}^{\tilde{g}}(x))}\leq C_{k,p,\epsilon}\cdot(r_{x})^{-\epsilon}

Therefore, for any $k\geq 1$ , $p>1$ and $\epsilon>0$ , there is some constant $C_{k,p,\epsilon}>0$ such that

(3.238)

|(\nabla^{t})^{k}\widetilde{\Phi}|_{W^{2,p}(B_{1/8}^{\tilde{g}}(x))}=|\Psi^{(k)}|_{W^{2,p}(B_{1/8}^{\tilde{g}}(x))}\leq C_{k,p,\epsilon}\cdot(r_{x})^{-\epsilon}.

This completes the proof of the claim.

Now we are in a position to finish the proof of the lemma by completing the induction arguments for Item (2). As before, for any $x$ , under the rescaled metric $\tilde{g}=(r_{x})^{-2}g$ , we start with the equation for $\widetilde{\Phi}$ in the rescaled geodesic ball $B_{1}^{\tilde{g}}(x)$ ,

(3.239)

\widetilde{\Delta}\widetilde{\Phi}+r_{x}\widetilde{Q}_{1}*\widetilde{\nabla}\widetilde{\Phi}+(r_{x})^{2}\widetilde{Q}_{2}*\widetilde{\Phi}=\tilde{\eta},\ 0<r_{x}<1,

where $\widetilde{Q}_{i}$ ’s are smooth functions and $|\tilde{\eta}|=O^{\prime}(1)$ . The above claim tells us that for any $k\geq 1$ , $p>1$ and $\epsilon>0$ ,

(3.240)

|(\nabla^{t})^{k}\widetilde{\Phi}|_{W^{2,p}(B_{1/2}^{\tilde{g}}(x))}\leq C_{k,p,\epsilon}\cdot(r_{x})^{-\epsilon},

where $C_{k,p,\epsilon}>0$ is independent of $x$ . Applying the Sobolev embedding, then for any $0<\alpha<1$ ,

(3.241)

|(\nabla^{t})^{k}\widetilde{\Phi}|_{C^{1,\alpha}(B_{1/4}^{\tilde{g}}(x))}\leq C_{k,\alpha,\epsilon}\cdot(r_{x})^{-\epsilon}.

In particular, $|(\nabla^{t})^{k}\widetilde{\Phi}|=O(r^{-\epsilon})$ for all $\epsilon>0$ . Notice that the above estimate is independent of the choice of $x$ . Rescaling back to the original metric, then for each $\epsilon>0$ ,

(3.242)

|(\nabla^{t})^{k}\nabla^{2}T|=O(r^{-\epsilon}).

The proof of the lemma is done.

∎

With all the above preparations, now we are ready to finish the proof of Theorem 3.10.

Proof of Theorem 3.10.

Let $\phi_{2}$ be the $3$ -current defined in Lemma 3.21 such that

(3.243)

\Delta\phi_{2}=2\pi\delta_{U}+\mathfrak{B}_{1},

with $\mathfrak{B}_{1}=O^{\prime}(1)$ . So Lemma 3.22 implies that there is some $3$ -current $\mathfrak{R}=O^{\prime}(r^{2})$ such that

(3.244)

\Delta\mathfrak{R}=\mathfrak{B}_{1}

and hence the $3$ -current

(3.245)

G_{U}\equiv\phi_{2}+\mathfrak{R}

satisfies the equation

(3.246)

\Delta G_{U}=2\pi\delta_{U}.

Moreover, by Lemma 3.21, $G_{U}$ has the expansion

	$\displaystyle G_{U}=$	$\displaystyle\frac{1}{2r}(1-\frac{H^{\alpha}y_{\alpha}}{2})dy_{1}\wedge dy_{2}\wedge dy_{3}+\frac{1}{2r}y_{\beta}A_{i\alpha\beta}dx_{i}\wedge dy_{\widehat{\alpha}}-\frac{1}{4}A_{ij\alpha\beta}r\cdot dy_{\widehat{\alpha\beta}}\wedge dx_{i}\wedge dx_{j}$
(3.247)			$\displaystyle+\frac{3}{16}(A_{i\alpha,\alpha+1}A_{j\alpha,\alpha+2}-A_{i\alpha,\alpha+2}A_{j\alpha,\alpha+1})d(ry_{\alpha})\wedge dx_{i}\wedge dx_{j}+r^{-3}\Pi_{3}^{(4)}+O^{\prime}(r^{2}).$

The proof of Theorem 3.10 is done.

∎

For our purpose later, we also need the following lemma.

Lemma 3.23.

Let $\Delta$ denote the Hodge Laplacian, then

(3.248)

\Delta(r^{2})=-6+\widetilde{O}(r^{2}).

Proof.

This follows from similar, and simpler arguments as above. First,

(3.249)

dr^{2}=2rdr=2y_{\alpha}\eta_{\alpha}+\widetilde{O}(r^{4}),

so it follows that

(3.250)

*dr^{2}=2y_{\alpha}\eta_{\widehat{\alpha}}\wedge\dvol_{T}+\widetilde{O}(r^{3})

and

(3.251)

d*dr^{2}=6\dvol_{N}\wedge\dvol_{T}+\widetilde{O}(r^{2})=6\dvol_{g}+\widetilde{O}(r^{2}).

Hence

(3.252)

\Delta(r^{2})=d^{*}dr^{2}=-*d*dr^{2}=-6+\widetilde{O}(r^{2}).

∎

3.3. Green’s currents on a cylinder

In this subsection we assume $Q\equiv D\times\mathbb{R}$ is a Riemannian product of a Kähler manifold $(D,\omega_{D},J_{D})$ of complex dimension $n-1$ , and the real line $\mathbb{R}$ with coordinate $z$ . Given a smooth divisor $H\subset D$ , let $P\equiv H\times\{0\}\subset D\times\{0\}$ . In our discussion sometimes we also naturally identify $H$ with $P$ . The results of this subsection will be purely local so $D$ and $H$ are not necessarily compact.

The splitting of a line $\mathbb{R}$ allows us to study the normal exponential map in $Q$ in terms of the normal exponential map in $D$ . Notice the normal bundle of $P$ in $Q$ is naturally a Riemannian direct sum

(3.253)

{N}=N_{0}\oplus\mathbb{R}_{z},

where $N_{0}$ is the normal bundle of $H$ in $D$ given as the orthogonal complement $(TH)^{\perp}$ of $TH$ in $TD|_{H}$ (with respect to $\omega_{D}$ ). So $N_{0}$ is naturally a hermitian line bundle. We also naturally identify $N_{0}$ with the holomorphic normal bundle $TD|_{H}/TH$ , as complex line bundles. Therefore, $N_{0}$ can be viewed as a holomorphic hermitian line bundle. The normal exponential map of $H$ in $D$ is defined by

(3.254)

\Exp_{H}:N_{0}\to D,\ (p,v)\mapsto\Exp_{p}(v),

which gives a local diffeomorphism from a neighborhood of the zero section in $N_{0}$ to a tubular neighborhood of $H$ in $D$ . Immediately,

(3.255)

d\text{Exp}_{H}:TN_{0}|_{H}\longrightarrow TD|_{H}

is the identity map under the natural isomorphisms $TN_{0}|_{H}\cong N_{0}\oplus TH$ and $TD|_{H}\cong N_{0}\oplus TH$ .

Given any point $p\in H$ , we may choose local holomorphic coordinates $\{w_{i}\}_{i=1}^{n-1}$ on $D$ , centered at $p$ , such that $H$ is locally defined by $w_{1}=0$ . Then $dw_{1},w_{2},\ldots,w_{n-1}$ induces local holomorphic coordinates on $N_{0}$ , which we denote by $\{\zeta,w_{2}^{\prime},\ldots,w_{n-1}^{\prime}\}$ . Given any $(p,v)\in N_{0}$ , its coordinates are by definition given as

(3.256)

\displaystyle\begin{cases}\zeta=(dw_{1})_{p}(v),\\ w_{j}^{\prime}=w_{j}(p),&j\geq 2.\end{cases}

Under the normal exponential map $\Exp_{H}$ , these coordinates can also be viewed as local (non-holomorphic) coordinates on $D$ , and when restricted to $H$ we have $w_{j}^{\prime}=w_{j}(j\geq 2)$ and $d\zeta=dw_{1}$ . In particular, $\{w_{2}^{\prime},\cdots,w_{n-1}^{\prime}\}$ still gives holomorphic coordinates on $H$ .

Similarly using $\Exp_{H}$ , the coordinate vector field $\partial_{\zeta}$ , originally defined on the normal bundle $N_{0}$ , can also be viewed as a local (non-holomorphic) vector field on $D$ . When restricted to $H$ , the vector field $\partial_{\zeta}$ can hence be identified with the local section $\sigma$ of $(TH)^{\perp}\subset TD|_{H}$ given by the orthogonal projection of the holomorphic vector field $\partial_{w_{1}}$ . Then we obtain a local unitary frame $e$ of $(TH)^{\perp}$ given by

(3.257)

e\equiv\sigma/|\sigma|.

These generate fiber coordinates $y,\bar{y}$ on $N_{0}$ such that

(3.258)

y=|\sigma|\cdot\zeta.

In this way we obtain local coordinates $\{y,\bar{y},y_{3}=z,w_{2}^{\prime},\bar{w}_{2}^{\prime},\ldots,w_{n-1}^{\prime},\bar{w}_{n-1}^{\prime}\}$ in a neighborhood of $p$ in $Q$ . To match with the notation in the previous subsection, with respect to the local orthonormal basis $\{\sigma+\bar{\sigma},\sqrt{-1}(\sigma-\bar{\sigma}),\partial_{z}\}$ , the normal geodesic coordinates are given by $\{y_{1}=Re(y),y_{2}=Im(y),y_{3}=z\}$ , and

(3.259)

r^{2}=|y|^{2}+z^{2}.

Also, the convention for orientation is given such that

(3.260)

2^{-(n-1)}\sqrt{-1}dy\wedge d\bar{y}\wedge dz\wedge\prod\limits_{j=2}^{n-1}(\sqrt{-1}dw_{j}^{\prime}\wedge d\bar{w}_{j}^{\prime})

defines a positive volume form. In the following, we will also use $\langle\cdot,\cdot\rangle$ to denote the hermitian inner product on $(1,0)$ -type vectors. The relation with the Riemannian inner product is seen as

(3.261)

\langle\xi,\xi\rangle=2\langle Re(\xi),Re(\xi)\rangle.

What the notation means will be clear in the context.

By making $P$ and $Q$ smaller we get local existence of Green’s current $G_{P}$ for $P$ in $Q$ , by Theorem 3.10, with the expansion given there. In our case the formula can be written in terms of the above complex coordinates

Proposition 3.24.

let $G_{P}$ be a Green’s current for $P\equiv H\times\{0\}$ in $Q$ , then locally

(3.262)

G_{P}=\psi\wedge dz+\mathcal{R},

where $\mathcal{R}\in C^{\infty}$ and $\psi$ is family of real-valued $(1,1)$ -forms on $D$ parametrized by $z$ , satisfying

(3.263)

\psi(-z)-\psi(z)\in C^{\infty}.

Moreover, in terms of the above local coordinates we can write

(3.264)

\psi=\frac{\sqrt{-1}}{4r}dy\wedge d\bar{y}+\frac{1}{2r}(yd\bar{y}+\bar{y}dy)\wedge\Gamma+r\cdot d\Gamma+r^{-3}\Pi_{2}^{(4)}+O^{\prime}(r^{2}),

where $\Gamma$ is a smooth real-valued $1$ -form locally defined on $H$ given by

(3.265)

\Gamma(v)\equiv-\frac{\sqrt{-1}}{2}\langle\nabla_{v}\partial_{y},\partial_{y}\rangle\Big|_{H},\ v\in T_{p}H,

and $\Pi_{2}^{(4)}$ is the $2$ -form given by Notation 3.9 such that it contains at least one of the $dy$ or $d\bar{y}$ .

Proof.

This essentially follows from the fact that $P$ is located on the slice $\{z=0\}$ and $H$ is a complex submanifold of $D$ . Indeed, we can decompose

(3.266)

G_{P}=\psi_{1}\wedge dz+\mathcal{R}_{1},

where $\mathcal{R}_{1}$ does not involve $dz$ . Given any compactly supported test form $\chi\in\Omega_{0}^{2n-4}(Q)$ , we can write

(3.267)

\chi=\beta\wedge dz+\gamma,

where $\gamma$ does not involve $dz$ . Immediately,

(3.268)

(\mathcal{R}_{1},\Delta\gamma)=\int_{Q}\mathcal{R}_{1}\wedge\Delta\gamma=0

and

(3.269)

(\psi_{1}\wedge dz,\Delta(\beta\wedge dz))=\int_{Q}\psi_{1}\wedge dz\wedge\Delta(\beta\wedge dz)=0.

So it follows that

(3.270)

(\mathcal{R}_{1},\Delta\chi)=(\mathcal{R}_{1},\Delta(\beta\wedge dz))=(G_{P},\Delta(\beta\wedge dz))=2\pi\int_{P}(\beta\wedge dz)=0.

This implies that $\Delta\mathcal{R}_{1}=0$ in the distributional sense. By the standard elliptic regularity, we have $\mathcal{R}_{1}\in C^{\infty}$ .

Now write

(3.271)

\psi_{1}=\psi+\psi_{2},

where $\psi$ is $J_{D}$ -invariant, i.e. of type $(1,1)$ in $D$ , and $\psi_{2}$ is anti- $J_{D}$ -invariant. Since $H$ is a complex submanifold of $D$ , the Dirac current $\delta_{P}$ is $J_{D}$ -invariant, hence $J_{D}(G_{P})$ is also a Green’s current for $P$ , so we see that $\psi_{2}=\frac{1}{2}(G_{P}-J(G_{P}))$ is smooth. Then we have

(3.272)

G_{P}=\psi\wedge dz+\mathcal{R},

where $\mathcal{R}=\mathcal{R}_{1}+\psi_{2}\wedge dz$ is smooth. Similarly since the $\delta_{P}$ is invariant under $z\mapsto-z$ , the difference $\psi(z)-\psi(-z)$ is smooth.

To see the expansion of $\psi$ , we notice that $H$ is a Kähler, in particular minimal, submanifold of $D$ . So the mean curvature of $H$ in $D$ vanishes. Also notice $\partial_{z}$ is parallel on $Q$ so $A_{i\alpha\beta}=0$ if either $\alpha=3$ or $\beta=3$ . This then implies that

(3.273)

\psi=\frac{\sqrt{-1}}{4r}dy\wedge d\bar{y}+\frac{1}{2r}(yd\bar{y}+\bar{y}dy)\wedge\Gamma+r\cdot\mathbb{A}+r^{-3}\Pi_{2}^{(4)}+\widetilde{O}(r^{2}),

where

(3.274)		$\displaystyle\Gamma$	$\displaystyle=-\frac{1}{2}A_{i12}dx_{i},$
(3.275)		$\displaystyle\mathbb{A}$	$\displaystyle=-\frac{1}{4}A_{ij\alpha\beta}dx_{i}\wedge dx_{j}.$

In particular, $\mathbb{A}=d\Gamma$ . Re-writing

(3.276)

A_{i12}=\langle\nabla_{\partial_{x_{i}}}\partial_{y_{2}},\partial_{y_{1}}\rangle

in terms of the complex coordinates $y,\bar{y}$ and bearing in mind (3.261) we obtain the desired formula for $\Gamma$ . ∎

Proposition 3.24 has a quick corollary which will be used in our later calculations.

Corollary 3.24.1.

For any positive integer $k\geq 2$ , we have

(3.277)

\psi^{k}=O^{\prime}(r^{k-1}).

Proof.

By Proposition 3.24, we write

(3.278)	$\displaystyle\psi$	$\displaystyle=\FT_{1}+\FT_{2}+\FT_{3},$
(3.279)	$\displaystyle\FT_{1}$	$\displaystyle\equiv\frac{\sqrt{-1}}{4r}dy\wedge d\bar{y},$
(3.280)	$\displaystyle\FT_{2}$	$\displaystyle\equiv\frac{1}{2r}(yd\bar{y}+\bar{y}dy)\wedge\Gamma=O^{\prime}(1),$
(3.281)	$\displaystyle\FT_{3}$	$\displaystyle\equiv r\cdot d\Gamma+r^{-3}\Pi_{2}^{(4)}+O^{\prime}(r^{2})=O^{\prime}(r).$

Immediately we have $(\FT_{1})^{2}=(\FT_{2})^{2}=\FT_{1}\wedge\FT_{2}=0$ and for all $k\geq 2$ , $\FT_{2}\wedge(\FT_{3})^{k}=(\FT_{3})^{k}=O^{\prime}(r^{k})$ . Moreover,

(3.282)

\FT_{1}\wedge\FT_{3}=\frac{\sqrt{-1}}{4}dy\wedge d\bar{y}\wedge d\Gamma+\frac{\sqrt{-1}}{4r^{4}}dy\wedge d\bar{y}\wedge\Pi_{2}^{(4)}+O^{\prime}(r).

Notice that $\frac{\sqrt{-1}}{4}dy\wedge d\bar{y}\wedge d\Gamma$ is a smooth term and by definition $dy\wedge d\bar{y}\wedge\Pi_{2}^{(4)}=0$ , then

(3.283)

\FT_{1}\wedge\FT_{3}=O^{\prime}(r).

So for all $k\geq 1$ ,

(3.284)

\FT_{1}\wedge(\FT_{3})^{k}=O^{\prime}(r^{k}).

Now by direct calculation,

(3.285)

\displaystyle\psi^{k}=k\Big((\FT_{1})\wedge(\FT_{3})^{k-1}+(\FT_{2})\wedge(\FT_{3})^{k-1}\Big)+(\FT_{3})^{k}.

The conclusion then follows. ∎

Notice that the above local coordinates $\{y,\bar{y}\}$ are not canonical, and depend on the initial choice of the local coordinates $\{w_{i}\}_{i=1}^{n-1}$ on $D$ . However, a different choice of local holomorphic coordinates on $D$ will induce the coordinates $\tilde{y},\bar{\tilde{y}}$ on fibers of $N_{0}$ such that

(3.286)

y=e^{\sqrt{-1}\phi}\cdot\tilde{y}

for some real function $\phi$ on $H$ . In particular, we have the transformation

(3.287)

dy\wedge d\bar{y}=d\tilde{y}\wedge d\bar{\tilde{y}}-\sqrt{-1}d|y|^{2}\wedge d\phi

and

(3.288)

\Gamma=\widetilde{\Gamma}-\frac{1}{2}d\phi,

This suggests viewing $\Gamma$ as a connection 1-form on the normal bundle. Indeed this is exactly the case.

Lemma 3.25.

$2\sqrt{-1}\Gamma$ is the Chern connection 1-form of the normal bundle $N_{0}$ with respect to the above hermitian holomorphic structure, in the local holomorphic frame $\sigma$ . In other words,

(3.289)

\Gamma=\frac{1}{2}d^{c}_{H}\log|\sigma|.

Proof.

By definition

(3.290)

\sigma=f\partial_{y}=\partial_{w_{1}}-\sum_{j\geq 2}\mu_{j}\partial_{w_{j}},

where $f=|\sigma|>0$ is local real valued function on $H$ , and $\mu_{2},\cdots,\mu_{n-1}$ are local complex valued function on $H$ . The key property we will use is that along $H$ , $\nabla_{\partial_{\bar{w}_{k}}}\sigma$ is tangential to $H$ for $k\geq 2$ . In fact, the Kähler condition implies $\nabla_{\bar{\partial}_{w_{k}}}\partial_{w_{j}}=0$ for all $j$ , and hence

(3.291)

\nabla_{\partial_{\bar{w}_{k}}}\sigma=\nabla_{\partial_{\bar{w}_{k}}}\Big(\partial_{w_{1}}-\sum_{j=2}^{n-1}\mu_{j}\partial_{w_{j}}\Big)=-\sum_{j=2}^{n-1}\partial_{\bar{w}_{k}}(\mu_{j})\partial_{w_{j}}.

Therefore,

(3.292)

\partial_{w_{k}}f=\partial_{w_{k}}\langle\partial_{y},\sigma\rangle=\langle\nabla_{\partial_{w_{k}}}\partial_{y},f\partial_{y}\rangle+\langle\partial_{y},\nabla_{\partial_{\bar{w}_{k}}}\sigma\rangle=f\langle\nabla_{\partial_{w_{k}}}\partial_{y},\partial_{y}\rangle,

and hence

(3.293)

\langle\nabla_{\partial_{w_{k}}}\partial_{y},\partial_{y}\rangle=f^{-1}\partial_{w_{k}}f=\partial_{w_{k}}(\log f).

Differentiating $|\partial_{y}|^{2}=1$ , we get

(3.294)

\langle\nabla_{\partial_{w_{k}}}\partial_{y},\partial_{y}\rangle+\langle\partial_{y},\nabla_{\partial_{\bar{w}_{k}}}\partial_{y}\rangle=0,

which implies

(3.295)

\langle\nabla_{\partial_{\bar{w}_{k}}}\partial_{y},\partial_{y}\rangle=-\partial_{\bar{w}_{k}}\log f.

Therefore,

	$\displaystyle\Gamma$	$\displaystyle=-\frac{\sqrt{-1}}{2}(\sum_{k\geq 2}\langle\nabla_{\partial_{w_{k}}}\partial_{y},\partial_{y}\rangle dw_{k}+\sum_{k\geq 2}\langle\nabla_{\partial_{\bar{w}_{k}}}\partial_{y},\partial_{y}\rangle d\bar{w}_{k})$
		$\displaystyle=-\frac{\sqrt{-1}}{2}(\sum_{k\geq 2}\partial_{w_{k}}(\log f)dw_{k}-\sum_{k\geq 2}\partial_{\bar{w}_{k}}(\log f)d\bar{w}_{k})$
		$\displaystyle=-\frac{\sqrt{-1}}{2}(\partial_{H}\log f-\bar{\partial}_{H}\log f)$
(3.296)			$\displaystyle=\frac{1}{2}d^{c}_{H}\log f.$

∎

For later applications we will need a few more local expansion results. We will also use the notation $O^{\prime}$ and $\widetilde{O}$ in Definition 3.3. The meaning is similar, but here we work on a neighborhood of $H$ in $D$ , and the distance function is locally given by $|y|$ . Notice the following expansions are given in the local (non-holomorphic) coordinates $\{y,\bar{y},w_{2}^{\prime},\bar{w}_{2}^{\prime},\cdots,w_{n-1},\bar{w}_{n-1}^{\prime}\}$ , and by definition we have $w_{j}^{\prime}|_{H}=w_{j}|_{H}$ for $j\geq 2$ .

Proposition 3.26.

The following holds locally near the point $p\in H$ ,

(3.297)

d^{c}_{D}|y|^{2}=\sqrt{-1}(yd\bar{y}-\bar{y}dy)+4|y|^{2}\cdot\Gamma+\widetilde{O}(|y|^{3}).

The proof relies on the following expansions of the holomorphic coordinate functions $w_{j}$ .

Lemma 3.27.

We have the expansion

(3.298)

\displaystyle\begin{cases}w_{1}=a_{1}y+a_{2}y^{2}+\widetilde{O}(|y|^{3})\\ w_{j}=w_{j}^{\prime}+c_{j}y+d_{j}y^{2}+\widetilde{O}(|y|^{3}),&j\geq 2,\end{cases}

where $a_{1}=|\sigma|^{-1}>0$ , $a_{2}$ , $c_{j}$ , $d_{j}$ are local smooth functions on $H$ .

Proof.

By definition, $\sigma$ is the orthogonal projection of $\partial_{w_{1}}$ onto $(TH)^{\perp}$ , so we have along $H$ ,

(3.299)

a_{1}^{-1}\partial_{y}=\partial_{w_{1}}+\sum_{j=2}^{n-1}b_{j}\partial_{w_{j}},

where $a_{1}=|\sigma|^{-1}>0$ and $b_{j}$ are smooth functions on $H$ . Now write

(3.300)

\partial_{y}=\sum_{j=1}^{n-1}\frac{\partial w_{j}}{\partial y}\partial_{w_{j}}+\sum_{j=1}^{n-1}\frac{\partial\bar{w}_{j}}{\partial{y}}\partial_{\bar{w}_{j}}.

then we get that along $H$ ,

(3.301)

\displaystyle\frac{\partial\bar{w}_{j}}{\partial y}=0,\ j\geq 1,

which in particular implies

(3.302)

\frac{\partial w_{j}}{\partial\bar{y}}=\overline{\frac{\partial\bar{w}_{j}}{\partial y}}=0,\ j\geq 1.

Now by the definition of the normal exponential map, we have at $p$ ,

(3.303)

\nabla_{\partial_{y}}\partial_{y}=\nabla_{\partial_{\bar{y}}}\partial_{y}=\nabla_{\partial_{\bar{y}}}\partial_{\bar{y}}=0.

Using the Kähler condition we have

(3.304)

\nabla_{\partial_{w_{j}}}{\partial_{\bar{w}_{k}}}=\nabla_{\partial_{\bar{w}_{j}}}\partial_{w_{k}}=0,\ \ j,k\geq 1.

Then by (3.300) we get

(3.305)

\frac{\partial^{2}w_{j}}{\partial y\partial\bar{y}}=\frac{\partial^{2}w_{j}}{\partial\bar{y}^{2}}=0,\ \ \ j\geq 1.

Therefore, the conclusion follows.

∎

Proof of Proposition 3.26.

Given the above Lemma we first obtain that

(3.306)

d\bar{w}_{1}=a_{1}d\bar{y}+\bar{y}(da_{1}+2\bar{a}_{2}d\bar{y})+\widetilde{O}(|y|^{2}),

then

(3.307)

w_{1}d\bar{w}_{1}=a_{1}^{2}yd\bar{y}+|y|^{2}a_{1}da_{1}+a_{1}y(2\bar{a}_{2}\bar{y}+a_{2}y)d\bar{y}+\widetilde{O}(|y|^{3}).

Hence

(3.308)

d_{D}^{c}|w_{1}|^{2}=\sqrt{-1}a_{1}^{2}(yd\bar{y}-\bar{y}dy)+\sqrt{-1}a_{1}\bar{a}_{2}\bar{y}(2yd\bar{y}-\bar{y}dy)-\sqrt{-1}a_{1}a_{2}y(2\bar{y}dy-yd\bar{y})+\widetilde{O}(|y|^{3}).

On the other hand, we have

(3.309)

|w_{1}|^{2}=a_{1}^{2}|y|^{2}+a_{1}(a_{2}y+\bar{a}_{2}\bar{y})|y|^{2}+\widetilde{O}(|y|^{4}).

(3.310)

d_{D}^{c}|w_{1}|^{2}=a_{1}^{2}d_{D}^{c}|y|^{2}+|y|^{2}d_{D}^{c}a_{1}^{2}+d_{D}^{c}(a_{1}(a_{2}y+\bar{a}_{2}\bar{y})|y|^{2})+\widetilde{O}(|y|^{3}).

Now by Lemma 3.27,

(3.311)

d_{D}^{c}(a_{1}y)=d_{D}^{c}w_{1}+\widetilde{O}(|y|)=-\sqrt{-1}dw_{1}+\widetilde{O}(|y|),

(3.312)

d_{D}^{c}y=-\sqrt{-1}dy+\widetilde{O}(|y|).

Similarly, $d_{D}^{c}\bar{y}=\sqrt{-1}d\bar{y}+\widetilde{O}(|y|)$ . Plugging these into (3.310), and compare with (3.308) we obtain

(3.313)

\displaystyle d_{D}^{c}|y|^{2}=\sqrt{-1}(yd\bar{y}-\bar{y}dy)-2|y|^{2}d_{D}^{c}\log a_{1}+\widetilde{O}(|y|^{3}).

Thanks to Lemma 3.27, $a_{1}=|\sigma|^{-1}$ which is a smooth function on $H$ , so

(3.314)

\displaystyle d_{D}^{c}|y|^{2}=\sqrt{-1}(yd\bar{y}-\bar{y}dy)+2|y|^{2}d_{H}^{c}\log|\sigma|+\widetilde{O}(|y|^{3}).

By Lemma 3.25, $\Gamma=\frac{1}{2}d_{H}^{c}\log|\sigma|$ , so we conclude

(3.315)

d_{D}^{c}|y|^{2}=\sqrt{-1}(yd\bar{y}-\bar{y}dy)+4|y|^{2}\Gamma+\widetilde{O}(|y|^{3}).

∎

Now we prove an expansion result for the trace of $\psi$ .

Proposition 3.28.

Let $\psi$ be the $2$ -form on $Q$ given as in (3.264), then we have the following expansion near $P$

(3.316)

\Tr_{\omega_{D}}\psi=\frac{1}{2r}+O^{\prime}(r).

Using (3.264) it is easy to see $\Tr_{\omega_{D}}\psi$ admits an expansion of the form

(3.317)

\Tr_{\omega_{D}}\psi=\frac{A_{0}}{r}+\frac{A_{1}y+\bar{A}_{1}\bar{y}}{r}+O^{\prime}(r).

for local functions $A_{0},A_{1}$ defined on $H$ . It suffices to show $A_{0}\equiv 1$ and $A_{1}\equiv 0$ . Since the left hand side is independent of the choice of local holomorphic coordinates, it suffices to we only need to work on the slice $z=0$ with special local holomorphic coordinates in a neighborhood of $p\in H$ , and it suffices to understand the Taylor expansion along the fiber $N_{0}(p)$ of $N_{0}$ over the fixed point $p$ .

Lemma 3.29.

We may choose the above holomorphic coordinates $\{w_{i}\}_{i=1}^{n-1}$ centered at $p$ , so that $H$ is given by $w_{1}=0$ and

(3.318)

\omega_{D}=\frac{\sqrt{-1}}{2}g_{i\bar{j}}dw_{i}\wedge d\bar{w}_{j},

where

(3.319)

\displaystyle\begin{cases}g_{i\bar{j}}(0)=\delta_{ij},&1\leq i,j\leq n-1,\\ \partial_{w_{1}}g_{i\bar{j}}(0)=0,&1\leq i,j\leq n-1,\\ \partial_{w_{k}}g_{1\bar{1}}(0)=\partial_{w_{k}}g_{i\bar{j}}(0)=0,&2\leq i,j,k\leq n-1.\end{cases}

Remark 3.29.1.

In fact, the only non-trivial Christoffel symbols at $p$ are

(3.320)

\Gamma_{ij}^{1}(0)=\partial_{i}g_{j\bar{1}}(0),\ \ \Gamma_{\bar{i}\bar{j}}^{\bar{1}}=\partial_{\bar{i}}g_{1\bar{j}}(0)

for $i,j\geq 2$ . This is due to the constraint that the equation $w_{1}=0$ defines $H$ , which prevents us from using substitutions like

(3.321)

w_{1}=z_{1}+\sum\limits_{i,j=2}^{n-1}C_{1ij}z_{i}z_{j}.

Intrinsically, $\{\Gamma^{1}_{ij}\}_{i,j\geq 2}$ captures the second fundamental form of the complex hypersurface $H$ at $p$ .

Proof of Lemma 3.29.

This follows from elementary manipulation. First, the holomorphic coordinates $\{w_{i}\}_{i=1}^{n-1}$ can be chosen such that $g_{i\bar{j}}(0)=\delta_{ij}$ for all $1\leq i,j\leq n-1$ . By the substitution of the form

(3.322)

\begin{cases}w_{i}=z_{i}+\frac{1}{2}\sum\limits_{j,k=2}^{n-1}C_{ijk}z_{j}z_{k}+\sum\limits_{j=2}^{n-1}D_{ij}z_{1}z_{j}+E_{i}z_{1}^{2},\ \ 2\leq i\leq n-1,\\ w_{1}=z_{1}+\sum\limits_{j=1}^{n-1}F_{j}z_{1}z_{j},\end{cases}

with suitable choices of coefficients, where $C_{ijk}=C_{ikj}$ for $2\leq i,j,k\leq n-1$ . One can plug both the Taylor expansion of $g_{i\bar{j}}$ along $z_{k}$ ’s and (3.322) into $\omega_{D}$ . Comparing the coefficients, then it follows that,

(3.323)

\displaystyle\begin{cases}C_{ijk}=-\partial_{w_{k}}g_{j\bar{i}}(0),\\ D_{ij}=-\partial_{w_{j}}g_{1\bar{i}}(0),\\ E_{i}=-\frac{1}{2}\partial_{w_{1}}g_{1\bar{j}}(0),\\ F_{i}=-\partial_{w_{j}}g_{1\bar{1}}(0),\\ F_{1}=-\frac{1}{2}\partial_{w_{1}}g_{1\bar{1}}(0),\end{cases}

where $2\leq i,j,k\leq n-1$ . Then we can achieve (3.319) with $\{w_{i}\}_{i=1}^{n-1}$ replaced by $\{z_{i}\}_{i=1}^{n-1}$ .

∎

Now we prove Proposition 3.28.

Proof of Proposition 3.28.

The goal is to show $A_{0}=1$ and $A_{1}=0$ in the expansion (3.317). We work in the above special coordinates centered at $p\in H$ .

The first step is to show that the $O^{\prime}(1)$ -term in the expansion of $\psi$ given by Proposition 3.24 in fact vanishes along $N_{0}(p)$ . To this end, notice that $\partial_{y}=\sigma=\partial_{w_{1}}$ at $p\in H$ and hence by Lemma 3.27,

(3.324)

w_{1}=y+a_{2}y^{2}+\widetilde{O}(|y|^{3}).

Since the only non-trivial Christofell symsbols at $p$ are $\Gamma_{ij}^{1}$ and $\Gamma_{\bar{i}\bar{j}}^{\bar{1}}$ for $i,j\geq 2$ , it easily follows that

(3.325)

d|\sigma|(p)=0.

Combining (3.325) and Lemma 3.25,

(3.326)

\Gamma(p)=\frac{1}{2}(d_{H}^{c}\log|\sigma|)(p)=0,

for each $p\in H$ . Therefore, along the fiber $N_{0}(p)$ of the normal bundle $N_{0}(p)$ , the expansion of $\psi$ in Proposition 3.24 becomes

(3.327)

\psi=\frac{\sqrt{-1}}{4|y|}dy\wedge d\bar{y}+O(|y|).

Next, we will compute the coefficients $A_{0}(p)$ and $A_{1}(p)$ in (3.317). As in the proof of Lemma 3.27, we obtain that

(3.328)

\frac{\partial w_{j}}{\partial y}(p)=\frac{\partial w_{j}}{\partial\bar{y}}=0,\ \ j\geq 2,

and

(3.329)

\frac{\partial^{2}w_{j}}{\partial y^{2}}(p)=\frac{\partial^{2}w_{j}}{\partial y\partial\bar{y}}(p)=\frac{\partial^{2}w_{j}}{\partial\bar{y}^{2}}(p)=0,\ \ j\geq 1.

This particularly implies that $a_{2}(p)=0$ and along the fiber $N_{0}(p)$ ,

(3.330)

w_{j}=O(|y|^{3}),\ \ j\geq 2.

By Lemma 3.29, $\partial_{w_{1}}g_{i\bar{j}}(p)=0$ for all $1\leq i,j\leq n-1$ , then the expansion of $\omega_{D}$ along the fiber $N_{0}(p)$ is at least quadratic in the $w_{1}$ -direction, i.e.

(3.331)		$\displaystyle\omega_{D}$	$\displaystyle=$	$\displaystyle\frac{\sqrt{-1}}{2}\Big(dw_{1}\wedge d\bar{w}_{1}+\sum_{j=2}^{n-1}dw_{j}\wedge d\bar{w}_{j}\Big)+O\Big(\sqrt{\sum_{j=2}^{n-1}\|w_{j}\|^{2}}\Big)+O(\|w_{1}\|^{2})$
(3.331)			$\displaystyle=$	$\displaystyle\frac{\sqrt{-1}}{2}(dw_{1}\wedge d\bar{w}_{1}+\sum_{j=2}^{n-1}dw_{j}\wedge d\bar{w}_{j})+O(\|y\|^{2}).$

By (3.324) and (3.330), along the fiber $N_{0}(p)$ , we have

(3.332)

dw_{1}=dy+O(|y|^{2}).

and

(3.333)

dw_{j}=dw_{j}^{\prime}+O(|y|^{2}),\ \ j\geq 2.

So we get

(3.334)

\omega_{D}=\frac{\sqrt{-1}}{2}(dy\wedge d\bar{y}+\sum_{j=2}^{n-1}dw_{j}^{\prime}\wedge d\bar{w}_{j}^{\prime})+O(|y|^{2})

Since by definition,

(3.335)

\Big(\Tr_{\omega_{D}}\psi\Big)\cdot\frac{\omega_{D}^{n-1}}{(n-1)!}=\psi\wedge\frac{\omega_{D}^{n-2}}{(n-2)!}.

by elementary manipulations we get that $A_{0}(p)=1$ and $A_{1}(p)=0$ . ∎

We close this subsection by proving an expansion of a local holomorphic volume form on $D$ . Given the choice of local holomorphic coordinates on $D$ as before, let $\Omega_{D}$ be a local holomorphic volume form in a neighborhood of $p$ , then we can always write

(3.336)

\Omega_{D}=f\cdot dw_{1}\wedge dw_{2}\cdots\wedge dw_{n-1},

for a local nowhere vanishing holomorphic function $f$ . Denote the local holomorphic volume form on $H$

(3.337)

\Omega_{H}\equiv dw_{2}^{\prime}\wedge\cdots dw_{n-1}^{\prime}=(dw_{2}\wedge\cdots\wedge dw_{n-1})|_{H}

Then $\Omega_{H}$ can be naturally viewed as a complex $(n-2)$ -form in some neighborhood of $p$ in $D$ , in the coordinate system given by $\{y,\bar{y},w_{2}^{\prime},\bar{w}_{2}^{\prime},\cdots,w_{n-1}^{\prime},\bar{w}_{n-1}^{\prime}\}$ .

Proposition 3.30.

We have the following expansion

(3.338)

\Omega_{D}=F(dy+2\sqrt{-1}y\Gamma)\wedge\Omega_{H}+\widetilde{O}(|y|)dy+\widetilde{O}(|y|^{2})

for some local smooth function $F$ on $H$ .

Proof.

We need to calculate the expansion for $dw_{1}\wedge\ldots\wedge dw_{n-1}$ . First, by Lemma 3.27,

(3.339)

dw_{1}=a_{1}(dy+yd_{H}\log a_{1})+\widetilde{O}(|y|)dy+\widetilde{O}(|y|^{2}),

where $a_{1}=|\sigma|^{-1}$ . Notice that

(3.340)

d_{H}\log a_{1}=2\partial_{H}\log a_{1}-\sqrt{-1}d_{H}^{c}\log a_{1}.

Applying Lemma 3.25,

(3.341)

d_{H}\log a_{1}=2(\partial_{H}\log a_{1}+\sqrt{-1}\Gamma).

Next, applying Lemma 3.27 to $w_{j}$ ’s for $j\geq 2$ ,

(3.342)

dw_{j}=dw_{j}^{\prime}+c_{j}dy+ydc_{j}+\widetilde{O}(|y|)dy+\widetilde{O}(|y|^{2}).

Since it holds that

(3.343)

\partial_{H}\log a_{1}\wedge\Omega_{H}\equiv 0,

then taking the wedge product,

(3.344)

dw_{1}\wedge\cdots\wedge dw_{n-1}=a_{1}(dy+2\sqrt{-1}y\Gamma)\wedge\Omega_{H}+\widetilde{O}(|y|)dy+\widetilde{O}(|y|^{2}).

On the other hand, we have the expansion of $f$ ,

(3.345)

f=f|_{H}+\frac{\partial f}{\partial y}|_{H}\cdot y+\frac{\partial f}{\partial\bar{y}}|_{H}\cdot\bar{y}+\widetilde{O}(|y|^{2}).

Therefore,

(3.346)

\Omega_{D}=f|_{H}a_{1}(dy+2\sqrt{-1}y\Gamma)\wedge\Omega_{H}+\widetilde{O}(|y|)dy+\widetilde{O}(|y|^{2}).

So we obtain the conclusion by taking $F=f|_{H}\cdot a_{1}$ .

∎

3.4. A global existence result

In this subsection, we will prove a global existence result for Green’s currents. Although the results hold in general Riemannian settings, for our application we shall only state the result in a special setting. We assume now $(D,\omega_{D},J_{D})$ is a compact Kähler manifold, $H$ is a smooth divisor Poincaré dual to $\frac{k}{2\pi}[\omega_{D}]$ for some positive $k\in\mathbb{Z}_{+}$ .

Proposition 3.31.

In the above context, given any constants $k_{-},k_{+}\in\mathbb{R}$ with

(3.347)

k_{-}-k_{+}=k,

there exists a unique global Green’s current $G_{P}$ for $P$ in $Q$ such that the following properties hold:

(1)

$G_{P}$ is of the form

(3.348) $G_{P}=\psi(z)\wedge dz.$

Moreover, for each $z\in\mathbb{R}$ , $\psi(z)$ is a closed real $(1,1)$ -current on $D$ .

(2)

For any nonnegative integer $k\in\mathbb{N}$ and for any $\delta\in(0,10^{-2})$ ,

(3.349)

\displaystyle\begin{cases}|\nabla^{k}(\psi(z)-(k_{-}z)\cdot\omega_{D})|=O(e^{(1-\delta)\sqrt{\lambda_{1}}z}),&z\rightarrow-\infty,\\ |\nabla^{k}(\psi(z)-(k_{+}z)\cdot\omega_{D})|=O(e^{-(1-\delta)\sqrt{\lambda_{1}}z}),&z\rightarrow\infty,\end{cases}

where $\lambda_{1}>0$ is the first eigenvalue of the Hodge Laplacian acting on closed real $(1,1)$ -forms on $D$ .

Remark 3.31.1.

This proposition can be seen as a generalization of theorem 2.6 in [HSVZ18] whose proof uses the general existence result of Green’s function on $\mathbb{T}^{2}\times\mathbb{R}$ . We thank Lorenzo Foscolo for discussions concerning the following proof via Fourier expansion, which is more constructive.

The proof of the proposition relies on the spectral analysis of the Hodge Laplacian. In particular, we need the following $C^{k}$ -estimate of the eigenforms in terms of the eigenvalues. This lemma will be also used in Section 5. The proof follows from standard $W^{2,p}$ -elliptic regularity and the Sobolev embedding theorems, so we omit it.

Lemma 3.32.

Let $(M^{m},g)$ be a closed Riemannian manifold of dimension $m\geq 2$ . For any $p\in\mathbb{N}$ , denote by $\Lambda^{(p)}\equiv\{\lambda_{j}\}_{j=0}^{\infty}$ with $\lambda_{0}=0$ the spectrum of the Hodge Laplacian $\Delta$ acting on the $p$ -forms. For any $k\in\mathbb{N}$ , there is some constant $C>0$ depending only on $(M,g)$ and $k$ , $p$ such that for all $\phi_{j}\in\Omega^{p}(M^{m})$ satisfying

(3.350)

\displaystyle\begin{cases}\Delta\phi_{j}=\lambda_{j}\phi_{j},\\ \|\phi_{j}\|_{L^{2}(M^{m})}=1,\end{cases}

we have

(3.351)

\|\nabla^{k}\phi_{j}\|_{C^{k}(M^{m})}\leq C\cdot(\lambda_{j})^{\frac{1}{2}[\frac{m}{2}]+\frac{k+1}{2}}.

Now we are ready to prove Proposition 3.31.

Proof of Proposition 3.31.

The uniqueness follows from the fact that the difference of any two Green’s currents for $P$ differ by a harmonic form, which must vanish by the asymptotic condition (3.349).

Now we focus on the proof of the global existence of $G_{P}$ on $Q$ . Let $\{\phi_{j}\}_{j=0}^{\infty}$ be a complete orthonormal basis of eigenvectors for the Hodge Laplacian acting on real-valued $2$ -forms on $D$ , and let $\Lambda_{D}\equiv\{\lambda_{j}\}_{j=0}^{\infty}$ be the corresponding spectrum. Our basic strategy is to first obtain a formal series expression of $G_{P}$ and then prove the convergence of this series.

To begin with, the Dirac $3$ -current $\delta_{P}$ of $P\subset Q$ has a formal expansion along the $D$ direction

(3.352)

\delta_{P}=\sum_{j=0}^{\infty}f_{j}(z)\phi_{j}\wedge dz,

where $f_{j}(z)$ is a $0$ -current on $\mathbb{R}$ and given by

(3.353)

f_{j}(z)\equiv 2\pi\Big(\int_{H}*_{D}\phi_{j}\Big)\delta_{0}(z),

where $\delta_{0}(z)$ is the standard Dirac $0$ -current acting on functions on $\mathbb{R}$ , supported at the slice $\{z=0\}$ . For $\lambda_{j}>0$ , by Hodge theory, $f_{j}$ is non-zero only when $\phi_{j}$ is a closed real $(1,1)$ -form because $H$ is a closed complex submanifold in $D$ . So we only restrict to the subset of such $\lambda_{j}$ ’s. Furthermore, if $\phi_{j}$ is harmonic, then

(3.354)

\int_{H}*_{D}\phi_{j}=\int_{D}\frac{d}{2\pi}\omega_{D}\wedge*_{D}\phi_{j}=\frac{d}{2\pi}\langle\omega_{D},\phi_{j}\rangle_{L^{2}(D)}.

It follows that there is exactly one $j$ , which we may assume to be $0$ , such that $\lambda_{j}=0$ and $f_{j}$ is non-zero. The corresponding eigenform is normalized to be

(3.355)

\phi_{0}\equiv\frac{1}{(\int_{D}\omega_{D}^{n}/n!)^{1/2}}\cdot\omega_{D}.

Now let $G_{P}$ be the formal series

(3.356)

G_{P}=\sum\limits_{j=0}^{\infty}h_{j}\cdot(\phi_{j}\wedge dz),

where $h_{j}$ satisfies

(3.357)

-h_{j}^{\prime\prime}(z)+\lambda_{j}\cdot h_{j}(z)=f_{j}(z).

For each $j\in\mathbb{Z}_{+}$ , we can write a formal solution

	$\displaystyle h_{j}(z)$	$\displaystyle=\frac{\pi}{-\sqrt{\lambda_{j}}}\Big(e^{-\sqrt{\lambda_{j}}\cdot z}\int_{-\infty}^{z}e^{\sqrt{\lambda_{j}}\cdot u}f_{j}(u)du+e^{\sqrt{\lambda_{j}}z}\int_{z}^{\infty}e^{-\sqrt{\lambda_{j}}\cdot u}f_{j}(u)du\Big)$
(3.358)			$\displaystyle=\begin{cases}\frac{\pi}{-\sqrt{\lambda_{j}}}\cdot e^{-\sqrt{\lambda_{j}}\cdot z}\cdot\int_{H}_{D}(\phi_{j}),&z>0,\\ \frac{\pi}{-\sqrt{\lambda_{j}}}\cdot e^{\sqrt{\lambda_{j}}\cdot z}\cdot\int_{H}_{D}(\phi_{j}),&z\leq 0.\end{cases}$

For $j=0$ , a solution is given by a piecewise linear function

(3.359)

\displaystyle h_{0}(z)=\begin{cases}k_{+}z,&z\geq 0,\\ k_{-}z,&z\leq 0.\end{cases}

Notice that the formal solution $h_{0}(z)$ is unique up to the addition of a linear function in $z$ . Fixing a choice of $h_{0}$ we then obtain a formal solution $G_{P}$ .

Next we show that the above formal series $G_{P}$ is well-defined by showing the formal solution indeed converges in the weak sense and has some exponential decaying rate as $|z|$ large, which consists of two steps.

In the first step, we claim that globally the formal expansion

(3.360)

G_{P}\equiv\sum\limits_{j=0}^{\infty}h_{j}\cdot(\phi_{j}\wedge dz)

in fact gives a well-defined $3$ -current on $Q$ and the series converges in the following sense: for any test form $\chi\in\Omega_{0}^{m-3}(Q)$ ,

(3.361)

\displaystyle\sum\limits_{j=0}^{N}\Big(h_{j}(z)\phi_{j}\wedge dz,\chi\Big)\longrightarrow(G_{P},\chi)\ \text{as}\ N\to\infty.

It suffices to show that for any smooth test form $\chi\in\Omega_{0}^{m-3}(Q)$ and for any $N\in\mathbb{N}$ ,

(3.362)

\sum\limits_{j=0}^{N}\Big|\Big(h_{j}(z)\phi_{j}\wedge dz,\chi\Big)\Big|\leq C_{0},

where $C_{0}>0$ is independent of $N$ . To see this, for each $j$ , we write

(3.363)		$\displaystyle\Big(h_{j}(z)\phi_{j}\wedge dz,\chi\Big)$	$\displaystyle=$	$\displaystyle\int_{Q}h_{j}(z)\phi_{j}\wedge dz\wedge\chi$
(3.363)			$\displaystyle=$	$\displaystyle\int_{\mathbb{R}}\Big(h_{j}(z)\cdot\int_{D}\langle\chi,*_{D}(\phi_{j})\rangle\Big)dz.$

The estimate (3.363) can be accomplished in the following manner. To begin with, we will show that the integral $\int_{D}\langle\chi,*_{D}(\phi_{j})\rangle$ has an uniform bound which is independent of $z$ . In fact, notice that $(\Delta)^{k}\phi_{j}=(\lambda_{j})^{k}\phi_{j}$ holds for any $k\in\mathbb{Z}_{+}$ , then

(3.364)	$\displaystyle\int_{D}\langle\chi(z),*_{D}(\phi_{j})\rangle\dvol_{g}$	$\displaystyle=$	$\displaystyle\frac{1}{(\lambda_{j})^{k}}\int_{D}\langle\chi(z),*_{D}(\Delta)^{k}(\phi_{j})\rangle\dvol_{g}$
		$\displaystyle=$	$\displaystyle\frac{1}{(\lambda_{j})^{k}}\int_{D}\langle\chi(z),(\Delta)^{k}*_{D}(\phi_{j})\rangle\dvol_{g}$
		$\displaystyle=$	$\displaystyle\frac{1}{(\lambda_{j})^{k}}\int_{D}\langle(\Delta)^{k}\chi(z),*_{D}(\phi_{j})\rangle.$

Lemma 3.32 implies

(3.365)

\|\phi_{j}\|_{C^{0}(D)}\leq C\cdot(\lambda_{j})^{\frac{n}{2}},

where $C$ depends only on $n$ and the metric $g$ . So it follows that

(3.366)

\Big|\int_{D}\langle\chi(z),*_{D}(\phi_{j})\rangle\dvol_{g}\Big|\leq C\cdot\|\chi\|_{C^{2k}(Q)}\cdot\frac{1}{(\lambda_{j})^{k-\frac{n}{2}}}.

Next, we will estimate the integral $\int_{\mathbb{R}}h_{j}(z)dz$ . To this end, for each $j\in\mathbb{Z}_{+}$ , let $\phi_{j}$ satsify

(3.367)

\displaystyle\begin{cases}\Delta\phi_{j}=\lambda_{j}\cdot\phi_{j}\\ \|\phi_{j}\|_{L^{2}(D)}=1.\end{cases}

By Lemma 3.32, for each $k\in\mathbb{N}$ ,

(3.368)

|\nabla^{k}_{\omega_{D}}\phi_{j}|\leq C(\lambda_{j})^{\frac{k+n}{2}}.

For fixed constant $z_{0}>10^{2}$ , applying (3.358) and (3.368),

(3.369)		$\displaystyle\Big\|\int_{\mathbb{R}}h_{j}(z)dz\Big\|$	$\displaystyle\leq$	$\displaystyle\int_{-\infty}^{-z_{0}}\|h_{j}(z)\|dz+\int_{-z_{0}}^{z_{0}}\|h_{j}(z)\|dz+\int_{z_{0}}^{+\infty}\|h_{j}(z)\|dz$
(3.369)			$\displaystyle\leq$	$\displaystyle C(1+\lambda_{j}^{\frac{n}{2}-1}).$

Combining the above estimates, we have

(3.370)

|(h_{j}(z)\phi_{j}\wedge dz,\chi)|=\Big|\int_{\mathbb{R}}\Big(h_{j}(z)\cdot\int_{D}\langle\chi,*_{D}(\phi_{j})\rangle\Big)dz\Big|\leq C\cdot\|\chi\|_{C^{2k}(Q)}\cdot\frac{C(1+\lambda_{j}^{\frac{n}{2}-1})}{(\lambda_{j})^{k-\frac{n}{2}}}.

Then applying Weyl’s law, if $k$ is sufficiently large, then the above series converges as stated in (3.362), which completes the proof of the claim.

At our next stage, we will study the exponential decaying behavior of the current $G_{P}$ defined in (3.360). For any $j\in\mathbb{Z}_{+}$ and for any number $z>10^{n^{2}+k^{2}}$ , we have

(3.371)

|\nabla^{k}_{Q}(h_{j}(z)\cdot\phi_{j})|\leq C(\lambda_{j})^{\frac{2n+k-1}{2}}e^{-\sqrt{\lambda_{j}}z}.

Notice that by elementary computations, for each $\delta\in(0,10^{-2})$ , there is some $z_{0}>10^{n^{2}+k^{2}}$ such that for all $z\in(z_{0},+\infty)$ and $j\in\mathbb{Z}_{+}$ ,

(3.372)

(\lambda_{j})^{\frac{2n+k-1}{2}}\cdot e^{-\delta\cdot\sqrt{\lambda_{j}}z}\leq(\lambda_{j})^{-6n}.

This implies that

(3.373)

\sum\limits_{j=0}^{\infty}|\nabla^{k}_{Q}(h_{j}(z)\cdot\phi_{j})|\leq Ce^{-(1-\delta)\sqrt{\lambda_{1}}z}\cdot\sum\limits_{j=0}^{\infty}(\lambda_{j})^{-6n}.

By Weyl’s law implies that the above numerical series converges, and hence for each $k\in\mathbb{N}$ $\nabla_{Q}^{k}(G_{P}-h_{0}(z))$ has an exponential decaying rate as $z\rightarrow+\infty$ . The argument is identical for $z<0$ .

The only remaining part is to show that the series $G_{P}$ defined by (3.360) satisfies the current equation

(3.374)

\Delta G_{P}=2\pi\delta_{P}

in the distributional sense, i.e., for any $\chi\in\Omega_{0}^{m-3}(Q)$ ,

(3.375)

(G_{P},\Delta\chi)=2\pi\int_{P}\chi.

Applying the definition of $f_{j}$ , $h_{j}$ and integration by parts, it is straightforward that for each $j\in\mathbb{N}$ ,

(3.376)

(h_{j}(z)\phi_{j}\wedge dz,\Delta\chi)=(f_{j}(z)\phi_{j}\wedge dz,\chi).

Since $\chi\in\Omega_{0}^{m-3}(Q)$ , the smooth $2$ -form $*_{D}(\chi)$ has the following $L^{2}$ -expansion on the slice $D\times\{0\}$ ,

(3.377)

*_{D}(\chi(0))=\sum\limits_{j=0}^{\infty}\langle*_{D}(\chi(0)),\phi_{j}\rangle_{D}\cdot\phi_{j}=\sum\limits_{j=0}^{\infty}\langle\chi(0),*_{D}(\phi_{j})\rangle_{D}\cdot\phi_{j}

and hence

(3.378)

\chi(0)=\sum\limits_{j=0}^{\infty}\Big(\int_{D}\phi_{j}\wedge\chi(0)\Big)*_{D}(\phi_{j}).

This implies that

(3.379)

\Big(\sum\limits_{j=0}^{\infty}f_{j}\cdot\phi_{j}\wedge dz,\chi\Big)=2\pi\sum\limits_{j=0}^{\infty}\Big(\int_{P}*_{D}(\phi_{j})\Big)\cdot\int_{D}\phi_{j}\wedge\chi(0)=\int_{P}\chi(0)=2\pi\int_{P}\chi.

Therefore,

(3.380)

(G_{P},\Delta\chi)=\Big(\sum\limits_{j=0}^{\infty}h_{j}\cdot\phi_{j}\wedge dz,\Delta\chi\Big)=\Big(\sum\limits_{j=0}^{\infty}f_{j}\cdot\phi_{j}\wedge dz,\chi\Big)=2\pi\int_{P}\chi=2\pi\delta_{P}(\chi),

which completes the proof. ∎

The constants $k_{-}$ and $k_{+}$ determines some information of the above $\psi$ .

Lemma 3.33.

Let $\psi$ be the $(1,1)$ -current in Proposition 3.31, then the following holds:

(1)

The cohomology class $[\partial_{z}\psi(z)]\in H^{2}(D;\mathbb{R})$ is given by $k_{-}[\omega_{D}]$ and $k_{+}[\omega_{D}]$ for $z<0$ and $z>0$ respectively.
(2)

At $z=0$ , we have

(3.381) $\partial_{z}\psi(0)=\frac{1}{2}(k_{-}+k_{+})\omega_{D}.$

In particular, it extends smoothly across $P$ .

Proof.

First, we prove Item (1). Since $Q$ is a Riemannian product, we have for $z\neq 0$ ,

(3.382)

\frac{d^{2}}{dz^{2}}\psi(z)=\Delta_{D}\psi(z)=dd^{*}\psi(z)

is exact, which implies that the cohomology class $[\psi(z)]\in H^{2}(D;\mathbb{R})$ is locally constant for $z\in\mathbb{R}\setminus\{0\}$ . On the other hand, by the exponential decay property in (3.349) we see that

(3.383)

\lim_{z\rightarrow\pm\infty}[\psi(z)]=\lim_{z\rightarrow\infty}k_{\pm}[\omega_{D}].

For Item (2), denote

(3.384)

\tilde{\psi}(z)\equiv\psi(-z)+(k_{-}+k_{+})z\omega_{D}.

Then $\tilde{\psi}\wedge dz$ is also a Green current for $P$ and it is also asymptotic to $k_{\pm}z\cdot\omega_{D}$ as $z\rightarrow\pm\infty$ . Therefore by uniqueness, $\tilde{\psi}(z)=\psi(z)$ . Taking the $z$ -derivative at $z=0$ we get the conclusion. ∎

(3.18)	$\displaystyle g_{\alpha\beta}\|_{(x,y)}$	$\displaystyle=\delta_{\alpha\beta}-\frac{1}{3}\Rm_{\alpha\gamma\xi\beta}\Big\|_{(x,0)}y_{\gamma}y_{\xi}+\widetilde{O}(r^{3}),$
(3.19)	$\displaystyle g_{ij}\|_{(x,y)}$	$\displaystyle=g^{P}_{ij}(x)+2\IIs^{\alpha}_{ij}\Big\|_{(x,0)}y_{\alpha}-(\Rm_{i\gamma\xi j}+\langle\nabla_{\partial_{x_{i}}}\partial_{y_{\gamma}},\nabla_{\partial_{x_{j}}}\partial_{y_{\xi}}\rangle)\Big\|_{(x,0)}y_{\gamma}y_{\xi}+\widetilde{O}(r^{3}),$
(3.20)	$\displaystyle g_{i\alpha}\|_{(x,y)}$	$\displaystyle=\langle\nabla_{\partial_{x_{i}}}\partial_{y_{\gamma}},\partial_{y_{\alpha}}\rangle\Big\|_{(x,0)}y_{\gamma}-\frac{2}{3}\Rm_{i\gamma\xi\alpha}\Big\|_{(x,0)}y_{\gamma}y_{\xi}+\widetilde{O}(r^{3}),$

3. Green’s currents [04ZC]

3. Green’s currents

3.1. Normal coordinates for an embedded submanifold

Definition 3.1 (Normal coordinates).

Lemma 3.2 (Generalized Gauss Lemma).

Definition 3.3 (Normal regularity order).

Lemma 3.4.

Proof.

3.2. Green’s currents for Riemannian submanifolds

Definition 3.5 (kk-current).

Definition 3.6 (Green’s current).

Example 3.7.

Proposition 3.8.

Proof.

Notation 3.9.

Remark 3.9.1.

Theorem 3.10.

Remark 3.10.1.

Remark 3.10.2.

Remark 3.10.3.

Remark 3.10.4.

Lemma 3.11.

Proof.

Lemma 3.12.

Proof.

Lemma 3.13.

Proof.

Lemma 3.14 (Rearrangement Lemma).

Proof.

Proposition 3.15.

Proof.

Lemma 3.16 (Cancellation Lemma).

Proof.

Lemma 3.17.

Proof.

Lemma 3.18.

Proof.

Lemma 3.19.

Proof.

Proposition 3.20.

Proof.

Lemma 3.21.

Proof.

Lemma 3.22.

Proof.

Proof of Theorem 3.10.

Lemma 3.23.

Proof.

3.3. Green’s currents on a cylinder

Proposition 3.24.

Proof.

Corollary 3.24.1.

Proof.

Lemma 3.25.

Proof.

Proposition 3.26.

Lemma 3.27.

Proof.

Proof of Proposition 3.26.

Proposition 3.28.

Lemma 3.29.

Remark 3.29.1.

Proof of Lemma 3.29.

Proof of Proposition 3.28.

Proposition 3.30.

Proof.

3.4. A global existence result

Proposition 3.31.

Remark 3.31.1.

Lemma 3.32.

Proof of Proposition 3.31.

Lemma 3.33.

Proof.

Definition 3.5 ( $k$ -current).