4.4. Structure near the singular locus I [045C]

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4.4. Structure near the singular locus I

There exists a constant $0<C\ll 1$ such that discs of $g_{a}$ -radius $CA^{1/4}$ centred at points in $S$ do not intersect each other; their union defines a disc bundle over $S$ : $\{R\lesssim A^{1/4}\}\subset(\mathbb{C}^{*})^{2}\times\mathbb{R}_{\mu}$ . We need to understand the local singularity structure of $v$ and $w^{p\bar{q}}$ in this disc bundle.

The following general setting is a variant of the classical Green’s function asymptote for submanifolds. Take the Euclidean space $(\mathbb{R}^{5},\sum_{i=1}^{5}ds_{i}^{2})$ , containing the codimension 3 graphical submanifold

\Gamma=\{(s_{1},s_{2},s_{3},s_{4},s_{5})|s_{i}=f_{i}(s_{1},s_{2}),i=3,4,5,|s_{1}|^{2}+|s_{2}|^{2}<1\}

such that $|f_{i}|_{C^{2}}\leq C$ for $i=3,4,5$ . Let $\vec{n}_{1},\vec{n}_{2},\vec{n}_{3}$ be an orthonormal frame on $\Gamma$ and define a local parametrisation of a tubular neighbourhood of $\Gamma$ :

\Psi(s_{1},s_{2},t_{1},t_{2},t_{3})=\vec{h}(s_{1},s_{2})+\sum_{1}^{3}t_{i}\vec{n}_{i},\quad\vec{h}(s_{1},s_{2})=(s_{1},s_{2},f_{i}(s_{1},s_{2}))\in\Gamma.

such that $\Gamma$ is $\{t_{1}=t_{2}=t_{3}=0\}$ in the coordinates $s_{1},s_{2},t_{1},t_{2},t_{3}$ . Denote $R=\sqrt{\sum t_{i}^{2}}$ as the Euclidean distance to $\Gamma$ . Let $f=f(\vec{h}(s_{1},s_{2}))$ be a function on $\Gamma$ with bound $\left\lVert f\right\rVert_{C^{2}}\leq C$ . We need asymptotes for the Green integral

I_{\Gamma,f}(s)=\int_{\Gamma}-\frac{1}{8\pi^{2}|s-s^{\prime}|^{3}}f(s^{\prime})d\text{Area}_{\Gamma}(s^{\prime}),\quad s\in\mathbb{R}^{5}.

We view $I_{\Gamma,f}$ as a function of $s_{1},s_{2},t_{1},t_{2},t_{3}$ .

Lemma 4.15.

For $\sum s_{i}^{2}\lesssim 1$ ,

\begin{cases}|I_{\Gamma,f}(s_{1},s_{2},t_{1},t_{2},t_{3})+\frac{f(s_{1},s_{2})}{4\pi R}|\leq C,\\ |\frac{\partial I_{\Gamma,f}}{\partial t_{i}}-\frac{f(s_{1},s_{2})t_{i}}{4\pi R^{3}}(1+\frac{1}{5}(\sum_{j}t_{j}\vec{n}_{j})\cdot\vec{H})|\leq C,\quad&i=3,4,5,\\ |\frac{\partial I_{\Gamma,f}}{\partial s_{i}}+\frac{1}{4\pi R}\frac{\partial f}{\partial s_{i}}(s_{1},s_{2})|\leq C,\quad&i=1,2.\end{cases}

where $\vec{H}$ is the mean curvature vector of $\Gamma$ at $\vec{h}(s_{1},s_{2})$ . If morever $\left\lVert f\right\rVert_{C^{3}}\leq C$ , $\left\lVert f_{i}\right\rVert_{C^{3}}\leq C$ , then

|\frac{\partial^{2}I_{\Gamma,f}}{\partial t_{i}\partial t_{j}}-\frac{(\delta_{ij}R^{2}-3t_{i}t_{j})f}{4\pi R^{5}}|\leq\frac{C}{R^{2}},\quad|\frac{\partial^{2}I_{\Gamma,f}}{\partial t_{i}\partial s_{j}}-\frac{t_{i}}{4\pi R^{3}}\frac{\partial f}{\partial s_{j}}|\leq\frac{C}{R},\quad|\frac{\partial^{2}I_{\Gamma,f}}{\partial s_{i}\partial s_{j}}|\leq\frac{C}{R}.

If morever $f(0)=0$ , then at $s_{1}=s_{2}=0$ ,

|\frac{\partial^{2}I_{\Gamma,f}}{\partial t_{i}\partial t_{j}}|\leq\frac{C}{R},\quad|\frac{\partial^{2}I_{\Gamma,f}}{\partial s_{i}\partial s_{j}}+\frac{1}{4\pi R}\frac{\partial^{2}f}{\partial s_{i}\partial s_{j}}|\leq C.

If morever $df(0)=0$ , then $|\frac{\partial^{2}I_{\Gamma,f}}{\partial t_{i}\partial s_{j}}|\leq C$ for $s_{1}=s_{2}=0$ .

Proof.

(Sketch) Consider $s_{1}=s_{2}=0$ . The leading order asymptote of $I_{\Gamma,f}$ is obtained by replacing $f$ with the constant $f(0)$ and replacing $\Gamma$ with $\mathbb{R}^{2}_{s_{1},s_{2}}$ . At $s=\sum t_{i}\vec{n}_{i}\in\mathbb{R}^{5}$ ,

\int_{\mathbb{R}^{2}}-\frac{1}{8\pi^{2}|s-s^{\prime}|^{3}}f(0)ds_{1}^{\prime}ds_{2}^{\prime}=-\frac{f(0)}{4\pi R}.

We then need to estimate the deviation of $I_{\Gamma,f}$ from this leading asymptote. After writing the surface integral as an integral over $s_{1},s_{2}$ plane, we reduce to the flat graph case $f_{3}=f_{4}=f_{5}=0$ . Writing

f(s^{\prime})=f(0)+\sum_{i=1}^{2}\frac{\partial f}{\partial s_{i}^{\prime}}(0)s_{i}^{\prime}+O(|s^{\prime}|^{2}),

we observe that the linear term does not contribute to $I_{\Gamma,f}(s)$ by parity, and the $O(|s^{\prime}|^{2})$ contribution is bounded by

C\int\frac{|s^{\prime}|^{2}}{|s-s^{\prime}|^{3}}ds_{1}^{\prime}ds_{2}^{\prime}\leq C\int\frac{r^{2}}{(r^{2}+R^{2})^{3/2}}rdr\leq C.

We now consider the normal first derivative $\frac{\partial I_{\Gamma,f}}{\partial t_{i}}$ for $s_{1}=s_{2}=0$ assuming without loss of generality that $df_{i}(0)=0$ . After using the Taylor expansion and parity trick above, modulo bounded terms

\begin{split}\frac{\partial I_{\Gamma,f}}{\partial t_{i}}\sim&f(0)\frac{3t_{i}}{8\pi^{2}}\int\frac{1}{(s_{1}^{\prime 2}+s_{2}^{\prime 2}+\sum_{j}(t_{j}-f_{j})^{2})^{5/2}}ds_{1}^{\prime}ds_{2}^{\prime}\\ \sim&f(0)\frac{3t_{i}}{8\pi^{2}}\int\frac{1}{(s_{1}^{\prime 2}+s_{2}^{\prime 2}+R^{2})^{5/2}}(1+\frac{2\sum t_{j}f_{j}}{s_{1}^{\prime 2}+s_{2}^{\prime 2}+R^{2}})ds_{1}^{\prime}ds_{2}^{\prime}\\ \sim&f(0)\frac{t_{i}}{4\pi R^{3}}(1+\frac{1}{5}\sum_{j}t_{j}(\frac{\partial^{2}f_{i}}{\partial s_{1}^{\prime 2}}+\frac{\partial^{2}f_{i}}{\partial s_{2}^{\prime 2}}))\\ =&f(0)\frac{t_{i}}{4\pi R^{3}}(1+\frac{1}{5}(\sum_{j}t_{j}\vec{n}_{j})\cdot\vec{H}),\end{split}

where $\vec{H}$ is the mean curvature of $\Gamma$ at the origin.

In the same setup, the tangential first derivative $\frac{\partial I_{\Gamma,f}}{\partial s_{i}}$ is modulo bounded terms

\begin{split}\frac{\partial I_{\Gamma,f}}{\partial s_{i}}\sim&\int f\frac{\partial}{\partial s_{i}^{\prime}}\frac{1}{8\pi^{2}|s-s^{\prime}|^{3}}ds_{1}^{\prime}ds_{2}^{\prime}\\ \sim&\int\frac{-1}{8\pi^{2}|s-s^{\prime}|^{3}}\frac{\partial f}{\partial s_{i}^{\prime}}ds_{1}^{\prime}ds_{2}^{\prime}\sim-\frac{1}{4\pi R}\frac{\partial f}{\partial s_{i}^{\prime}}(0).\end{split}

The argument for second derivatives are similar. ∎

Around a point $P\in S$ of interest, we introduce linear change of coordinates,

\begin{cases}\xi^{\prime}_{1}=\frac{z_{1}(P)(\eta_{1}-\eta_{1}(P))+z_{2}(P)(\eta_{2}-\eta_{2}(P))}{A^{1/2}|a^{i\bar{j}}z_{i}(P)\bar{z}_{j}(P)|^{1/2}},\\ \xi_{2}=\frac{(a^{1\bar{1}}\bar{z}_{1}(P)+a^{1\bar{2}}\bar{z}_{2}(P))(\eta_{2}-\eta_{2}(P))-(a^{2\bar{1}}\bar{z}_{1}(P)+a^{2\bar{2}}\bar{z}_{2}(P))(\eta_{1}-\eta_{1}(P)))}{|a^{i\bar{j}}z_{i}(P)\bar{z}_{j}(P)|^{1/2}},\end{cases}

such that $T_{P}S=\{\xi^{\prime}_{1}=0,\mu=0\}$ , and $\xi_{2}=0$ on the normal 3-plane to $T_{P}S$ . In these new linear coordinates,

g_{a}=A(d\mu^{2}+|d\xi^{\prime}_{1}|^{2}+|d\xi_{2}|^{2}),\quad d\eta_{1}\wedge d\eta_{2}=A^{1/2}d\xi_{1}^{\prime}\wedge d\xi_{2}.

A problem is that the tangent planes tilts as $P$ moves along $S$ . We find a local smooth vector valued function $\vec{h}$ to represent $S$ locally as a graph

S\cap\{|\xi_{2}|\lesssim A^{-1/4}\}=\{(\xi_{1},\xi_{2},\mu)=\vec{h}(\xi_{2}),|\xi_{2}|\lesssim A^{-1/4}\}\subset(\mathbb{C}^{*})^{2}\times\mathbb{R}_{\mu}.

The $g_{a}$ -normal (1,0)-type vector to $S\subset(\mathbb{C}^{*})_{\eta_{1},\eta_{2}}^{2}$ at the point $\vec{h}(\xi_{2})\in S$ is

\vec{n}(\vec{h}(\xi_{2}))=\frac{a^{p\bar{q}}\bar{z}_{q}}{|a^{i\bar{j}}z_{i}\bar{z}_{j}|^{1/2}}(\vec{h}(\xi_{2}))\frac{\partial}{\partial\eta_{p}}.

Denote $r=\sqrt{A(|\xi_{1}|^{2}+|\xi_{2}|^{2}+\mu^{2})}$ . Define a local diffeomorphism $\Psi$ on the local chart $\{r\lesssim A^{1/4}\}$ ,

\Psi(\xi_{1},\xi_{2},\mu)=\vec{h}(\xi_{2})+A^{1/2}\xi_{1}\vec{n}(\vec{h}(\xi_{2}))+\mu\vec{e}_{\mu}\in(S^{1}\times\mathbb{R})_{\eta_{1},\eta_{2}}^{2}\times\mathbb{R}_{\mu},

where $\vec{e}_{\mu}$ denotes the basis vector with $g_{a}$ -magnitude $A^{1/2}$ corresponding to the $\mu$ -variable. The image of $\Psi$ is a tubular neighbourhood of a graphical subset of $S$ , and the straight degeneracy locus $\{\mu=\xi_{1}=0\}$ is identified with the curved degeneracy locus $S$ . The pullback function $\Psi^{*}R=A^{1/2}\sqrt{\mu^{2}+|\xi_{1}|^{2}}$ , and $\Psi^{*}g_{a}$ satisfies

(4.19)

\begin{cases}|\Psi^{*}g_{a}-A(|d\xi_{1}|^{2}+|d\xi_{2}|^{2}+d\mu^{2})|_{g_{a}}\leq CA^{1/4}|\xi_{2}|,\\ |\nabla^{k}_{g_{a}}\{\Psi^{*}g_{a}-A(|d\xi_{1}|^{2}+|d\xi_{2}|^{2}+d\mu^{2})\}|_{g_{a}}\leq CA^{-k/4},\quad k\geq 1.\end{cases}

Proposition 4.16.

(leading order asymptote near $S$ ) Via the local diffeomorphism $\Psi$ , on the chart $\{r\lesssim A^{1/4}\}$ ,

\begin{cases}|\Psi^{*}(w^{p\bar{q}}d\eta_{p}\otimes d\bar{\eta}_{q})-\frac{1}{2\sqrt{\mu^{2}+|\xi_{1}|^{2}}}d\xi_{1}\otimes d\bar{\xi}_{1}|_{g_{a}}\leq CA^{-3/4}\max(1,\log(A^{-1/4}\varrho),\\ |\Psi^{*}v-\frac{1}{2\sqrt{\mu^{2}+|\xi_{1}|^{2}}}|\leq CA^{1/4}\max(1,\log(A^{-1/4}\varrho),\end{cases}

Remark 4.4.

In the original coordinates, for $R\lesssim A^{1/4}$ ,

\begin{cases}|w^{p\bar{q}}-\frac{z_{p}\bar{z}_{q}}{2RA^{1/2}a^{i\bar{j}}z_{i}\bar{z}_{j}}|\leq CA^{-1/4}\max(1,\log(A^{-1/4}\varrho)),\\ |v-\frac{A^{1/2}}{2R}|\leq CA^{1/4}\max(1,\log(A^{-1/4}\varrho)).\end{cases}

Proof.

We focus on $w^{1\bar{1}}=\gamma_{1}+\gamma_{3}$ , where $\gamma_{1},\gamma_{3}$ admit the Green’s representation (4.11). We split the integral on $S$ into the short distance contribution from $S\cap\{|\eta_{1}-\eta^{\prime}_{1}|,|\eta_{2}-\eta^{\prime}_{2}|\lesssim\frac{1}{2}\}$ and the long distance contribution from $S\cap\{|\eta_{1}-\eta^{\prime}_{1}|\gtrsim\frac{1}{2}\text{ or }|\eta_{2}-\eta^{\prime}_{2}|\gtrsim\frac{1}{2}\}$ .

The short distance contribution to the integral $w^{1\bar{1}}$ is

-\pi A^{1/2}\int_{S\cap\{|\eta_{1}-\eta^{\prime}_{1}|,|\eta_{2}-\eta^{\prime}_{2}|\lesssim\frac{1}{2}\}}\gamma(\eta_{1}-\eta_{1}^{\prime},\eta_{2}-\eta_{2}^{\prime},\mu)\sqrt{-1}d\eta_{2}^{\prime}\wedge d\bar{\eta}_{2}^{\prime}.

Applying Lemma 4.2, we can replace the periodic Newtontian potential $\gamma$ by the ordinary Newtonian potential, so the short distance contribution is replaced by

-\pi A^{1/2}\int_{S\cap\{|\eta_{1}-\eta^{\prime}_{1}|,|\eta_{2}-\eta^{\prime}_{2}|\lesssim\frac{1}{2}\}}-\frac{1}{8\pi^{2}|(\eta_{1}-\eta_{1}^{\prime},\eta_{2}-\eta_{2}^{\prime},\mu)|_{a}^{3}}\sqrt{-1}d\eta_{2}^{\prime}\wedge d\bar{\eta}_{2}^{\prime},

at a cost of a smooth error of order $O(A^{-1/4})$ . The measure $\sqrt{-1}d\eta_{2}^{\prime}\wedge d\bar{\eta}_{2}^{\prime}$ is equal to $2\frac{|z_{1}^{\prime}|^{2}}{Aa^{i\bar{j}}z_{i}^{\prime}\bar{z}_{j}^{\prime}}d\mathcal{A}(\eta_{1}^{\prime},\eta_{2}^{\prime})$ , so the above expression is

\int_{S\cap\{|\eta_{1}-\eta^{\prime}_{1}|,|\eta_{2}-\eta^{\prime}_{2}|\lesssim\frac{1}{2}\}}-\frac{1}{8\pi^{2}|(\eta_{1}-\eta_{1}^{\prime},\eta_{2}-\eta_{2}^{\prime},\mu)|_{a}^{3}}\frac{-2\pi|z_{1}^{\prime}|^{2}}{A^{1/2}a^{i\bar{j}}z_{i}^{\prime}\bar{z}_{j}^{\prime}}d\mathcal{A}(\eta_{1}^{\prime},\eta_{2}^{\prime}).

We may assume the submanifold $S\cap\{|\eta_{1}-\eta^{\prime}_{1}|,|\eta_{2}-\eta^{\prime}_{2}|\lesssim\frac{1}{2}\}$ is graphical, so Lemma 4.15 applies after scaling. Thus the short distance contribution to $w^{1\bar{1}}$ is

-\frac{1}{4\pi R}\frac{-2\pi|z_{1}^{\prime}|^{2}}{A^{1/2}a^{i\bar{j}}z_{i}^{\prime}\bar{z}_{j}^{\prime}}+O(A^{-1/4})=\frac{|z_{1}^{\prime}|^{2}}{2RA^{1/2}a^{i\bar{j}}z_{i}^{\prime}\bar{z}_{j}^{\prime}}+O(A^{-1/4}),

where the complex coordinates $z_{i}^{\prime}$ are computed at $\vec{h}(\xi_{2})\in S$ . But the factor $\frac{|z_{1}|^{2}}{A^{1/2}a^{i\bar{j}}z_{i}\bar{z}_{j}}$ varies slowly, so we may as well compute it at $(\eta_{1},\eta_{2},\mu)$ .

The long distance contribution to $w^{1\bar{1}}$ is $O(A^{-1/4}\max(1,\log(A^{-1/4}\varrho)))$ by following the same steps as in Section 4.2, 4.3, using Lemma 4.1. Combining the two contributions,

|w^{1\bar{1}}-\frac{|z_{1}|^{2}}{2RA^{1/2}a^{i\bar{j}}z_{i}\bar{z}_{j}}|\leq CA^{-1/4}\max(1,\log(A^{-1/4}\varrho))).

The cases of $w^{p\bar{q}}$ and $v=Aa^{p\bar{q}}w^{p\bar{q}}$ are similar. ∎

Corollary 4.17.

Fix $0<\epsilon_{0}\ll 1$ , then on the total space

M^{-}=\{A^{-1/4}\varrho<\exp(\epsilon_{0}A^{3/4})\},

the function $V_{(1)}$ is positive and the matrix $W^{p\bar{q}}_{(1)}$ is positive definite.