6. Perturbation to Calabi-Yau metrics on the neck [054R]

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6. Perturbation to Calabi-Yau metrics on the neck

In Section 4.1 we have constructed a family of $C^{2,\alpha}$ -Kähler structures $(\omega_{T},\Omega_{T})$ on $\mathcal{M}_{T}$ with weighted error estimate by Proposition 4.23. Our goal in this Section is to perturb $\omega_{T}$ to a genuine Calabi-Yau metric for $T$ sufficiently large. This amounts to applying the quantitative implicit function theorem (Lemma 6.1). The main result is Theorem 6.3. In Section 6.4 we also compute the measured Gromov-Hausdorff limit of these metrics at an appropriate scale. As mentioned in the Introduction, it is the proof, but not Theorem 1.1 itself, that will be immediately used in the proof of Theorem 1.1.

6.1. Framework of perturbation

The studies and applications of the implicit function theorem have been well developed in various contexts. We refer the readers to the book [KP13] for seeing the comprehensive discussions and the history of the whole methodology. For our practical and specific applications, we need the following quantitative version of implicit function theorem (Lemma 6.1), which is based on Banach contraction mapping principle.

To avoid confusions, we clarify several notations as follows:

•

Let $\mathscr{L}:\mathfrak{A}\to\mathfrak{B}$ be a bounded linear operator between normed linear spaces $\mathfrak{A}$ and $\mathfrak{B}$ , then the operator norm of $\mathscr{L}$ is defined by

(6.1)

\|\mathscr{L}\|_{op}\equiv\inf\Big\{M_{0}\in\mathbb{R}_{+}\Big|\ \|\mathscr{L}(\bm{v})\|_{\mathfrak{B}}\leq M_{0}\cdot\|\bm{v}\|_{\mathfrak{A}},\ \forall\bm{v}\in\mathfrak{A}\Big\}.

•

We use the common notation $\bm{0}$ for the zero vector in every normed linear space.

Lemma 6.1 (Implicit function theorem).

Let $\mathscr{F}:\mathfrak{A}\to\mathfrak{B}$ be a map between two Banach spaces such that for all $\bm{v}\in\mathfrak{A}$ ,

(6.2)

\mathscr{F}(\bm{v})-\mathscr{F}(\bm{0})=\mathscr{L}(\bm{v})+\mathscr{N}(\bm{v}),

where the operator $\mathscr{L}:\mathfrak{A}\to\mathfrak{B}$ is linear and the operator $\mathscr{N}:\mathfrak{A}\to\mathfrak{B}$ satisfies $\mathscr{N}(\bm{0})=\bm{0}$ . Additionally we assume the following properties:

(1)

(Bounded inverse) $\mathscr{L}:\mathfrak{A}\to\mathfrak{B}$ is an isomorphism and there is some constant $C_{L}>0$ such that

(6.3) $\|\mathscr{L}^{-1}\|_{op}\leq C_{L},$

where $\mathscr{L}^{-1}$ is the inverse of $\mathscr{L}$ .

(2)

There exists a constant $C_{N}>0$ and there is some $r_{0}\in(0,\frac{1}{2C_{L}C_{N}})$ satisfying the following:

(a)

(Controlled nonlinear error) for all $\bm{v}_{1},\bm{v}_{2}\in\overline{B_{r_{0}}(\bm{0})}\subset\mathfrak{A}$ ,

(6.4)

\|\mathscr{N}(\bm{v}_{1})-\mathscr{N}(\bm{v}_{2})\|_{\mathfrak{B}}\leq C_{N}\cdot r_{0}\cdot\|\bm{v}_{1}-\bm{v}_{2}\|_{\mathfrak{A}}.

(b)

(Controlled initial error) $\mathscr{F}(\bm{0})$ is effectively controlled as follows,

(6.5) $\|\mathscr{F}(\bm{0})\|_{\mathfrak{B}}\leq\frac{r_{0}}{4C_{L}}.$

Then the equation $\mathscr{F}(\bm{x})=\bm{0}$ has a unique solution $\bm{x}\in B_{r_{0}}(\bm{0})$ with the estimate

(6.6)

\|\bm{x}\|_{\mathfrak{A}}\leq 2C_{L}\cdot\|\mathscr{F}(\bm{0})\|_{\mathfrak{B}}.

Remark 6.1.1.

In our applications, the constants $C_{L}>0$ , $C_{N}>0$ and $r_{0}>0$ will be fixed as uniform constants (independent of $T\gg 1$ ). We will see this from the global linear and nonlinear estimates, which will be stated and proved in next subsections. With the specified weight parameters $\delta,\mu,\nu$ , the error estimate in Proposition 4.23 in fact guarantees $\|\mathrm{Err}_{CY}\|_{\mathfrak{B}}\to 0$ as $T\to\infty$ , which particularly implies $\|\mathscr{F}(\bm{0})\|_{\mathfrak{B}}\to 0$ and hence $\mathscr{F}$ satisfies (b) of Item (2) in the above lemma.

To set up the perturbation problem in our setting, we define the Banach spaces

	$\displaystyle\mathfrak{S}_{1}$	$\displaystyle\equiv\Big\{\sqrt{-1}\partial\bar{\partial}\phi\in\Omega^{1,1}(\mathcal{M}_{T})\Big\|\phi\in C^{2,\alpha}(\mathcal{M}_{T})\ \text{is}\ S^{1}\text{-invariant and satisfies}\ \frac{\partial\phi}{\partial n}\Big\|_{\partial\mathcal{M}_{T}}=0\Big\},$
(6.7)		$\displaystyle\mathfrak{S}_{2}$	$\displaystyle\equiv\Big\{f\in C^{0,\alpha}(\mathcal{M}_{T})\Big\|f\ \text{is}\ S^{1}\text{-invariant and}\ \int_{\mathcal{M}_{T}}f\cdot\omega_{T}^{n}=0\Big\}.$

endowed with the weighted Hölder norms

(6.8)		$\displaystyle\\|\sqrt{-1}\partial\bar{\partial}\phi\\|_{\mathfrak{S}_{1}}$	$\displaystyle\equiv\\|\sqrt{-1}\partial\bar{\partial}\phi\\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(X_{t})},$
(6.9)		$\displaystyle\\|f\\|_{\mathfrak{S}_{2}}$	$\displaystyle\equiv\\|\sqrt{-1}\partial\bar{\partial}\phi\\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(X_{t})}.$

Notice an $S^{1}$ invariant function $\phi$ on $\mathcal{M}_{T}$ can be identified with a function on the quotient $Q_{T}$ , and the Neumann boundary condition $\frac{\partial\phi}{\partial n}|_{\partial\mathcal{M}_{T}}=0$ amounts to the condition $\partial_{z}\phi=0$ on $\partial Q_{T}$ .

In this section, the weight parameters are specified as follows:

(NP1)

(Fix $\nu$ ) The parameters $\nu\in\mathbb{R}$ is chosen such that

(6.10) $\displaystyle\nu\in(-1,0).$

In our context, Lemma 6.5 requires $\nu\in(-1,1)$ . To effectively apply Proposition 4.23, we need $\nu\in(-1,0)$ .
(NP2)

(Fix $\alpha$ ) The Hölder order $\alpha\in(0,1)$ is chosen sufficiently small such that

(6.11) $\nu+\alpha<0.$
(NP3)

(Fix $\delta$ ) $\delta>0$ is chosen such that

(6.12) $0<\delta<\delta_{N}\equiv\frac{1}{n\cdot(|k_{-}|+|k_{+}|)^{n}}\cdot\min\{\delta_{b},\delta_{e},\sqrt{\lambda_{D}}\},$

where $\epsilon_{X}>0$ is the constant in Theorem 5.2, $\sqrt{\lambda_{D}}$ is in Lemma 6.7 (Liouville theorem on $Q$ ), $\delta_{b}>0$ is in Proposition 5.14 (Liouville theorem on the Calabi space $\mathcal{C}^{n}$ around the boundary of the neck), $\delta_{e}>0$ is in the error estimate Proposition 4.23.
(NP4)

(Fix $\mu$ ) The parameter $\mu$ is fixed by

(6.13) $\mu=(1-\frac{1}{n})(\nu+2+\alpha).$

This condition guarantees that the weight function $\rho_{\delta,\nu,\mu}^{(\alpha)}$ with parameters specified as the above is uniformly bounded from below. This will be used in proving Proposition 6.4.

We first normalize the holomorphic volume form. For $T\gg 1$ , starting with the $C^{2,\alpha}$ -Kähler structure $(\omega_{T},\Omega_{T})$ , we will solve the Calabi-Yau equation

(6.14)

\frac{1}{n!}(\omega_{T}+\sqrt{-1}\partial\bar{\partial}\phi)^{n}=(\sqrt{-1})^{n^{2}}2^{-n}\cdot\Omega_{T}\wedge.\bar{\Omega}_{T}.

Notice that

(6.15)

\displaystyle\int_{\mathcal{M}_{T}}\frac{(\sqrt{-1})^{n^{2}}}{2^{n}}\Omega_{T}\wedge\bar{\Omega}_{T}

\displaystyle=\int_{\mathcal{M}_{T}}h\cdot\frac{\omega_{D}^{n-1}}{(n-1)!}dz\wedge\Theta=\int_{T_{-}}^{T_{+}}dz\int_{D}h\frac{\omega_{D}^{n-1}}{(n-1)!}

and

(6.16)

\displaystyle\int_{\mathcal{M}_{T}}\frac{\omega_{T}^{n}}{n!}=T^{2-n}\int_{\mathcal{M}_{T}}\frac{\tilde{\omega}(z)^{n-1}}{(n-1)!}dz\wedge\Theta=\frac{T^{2-n}}{(n-1)!}\int_{T_{-}}^{T_{+}}dz\int_{D}\tilde{\omega}(z)^{n-1}=C_{1}T^{2}

for some computable constant $C_{1}>0$ . So by (4.14) we get

(6.17)

\int_{\mathcal{M}_{T}}\frac{(\sqrt{-1})^{n^{2}}}{2^{n}}\Omega_{T}\wedge\bar{\Omega}_{T}=(1+O(T^{-2}))\int_{\mathcal{M}_{T}}\frac{\omega_{T}^{n}}{n!}.

Now we replace $\Omega_{T}$ by

(6.18)

\Big(\frac{\int_{\mathcal{M}_{T}}\frac{(\sqrt{-1})^{n^{2}}}{2^{n}}\Omega_{T}\wedge\bar{\Omega_{T}}}{\int_{\mathcal{M}_{T}}\frac{\omega_{T}^{n}}{n!}}\Big)^{-\frac{1}{2}}\Omega_{T}

Then we have

(6.19)

\frac{(\sqrt{-1})^{n^{2}}}{2^{n}}\Omega_{T}\wedge\bar{\Omega}_{T}=(1+\mathrm{Err}_{CY})\frac{\omega_{T}^{n}}{n!}

where

(6.20)

\int_{\mathcal{M}_{T}}\mathrm{Err}_{CY}\omega_{T}^{n}=0.

Applying Proposition 4.23 and (6.13),

(6.21)

\|\mathrm{Err}_{CY}\|_{C^{0,\alpha}_{\delta,\nu+2,\mu}(\mathcal{M}_{T})}=O(T^{\nu+\alpha}).

Let $\mathscr{F}$ be the map sending every $\sqrt{-1}\partial\bar{\partial}\phi\in\mathfrak{S}_{1}$ to the function which satisfies

(6.22)

\mathscr{F}(\sqrt{-1}\partial\bar{\partial}\phi)\cdot\omega_{T}^{n}\equiv(\omega_{T}+\sqrt{-1}\partial\bar{\partial}\phi)^{n}-\omega_{T}^{n}(1-\mathrm{Err}_{CY}).

Then (6.21) immediately tells us that

(6.23)

\|\mathscr{F}(\bm{0})\|_{\mathfrak{S}_{2}}=O(T^{\nu+\alpha}).

Lemma 6.2.

$\mathscr{F}(\mathfrak{S}_{1})\subset\mathfrak{S}_{2}$ .

Proof.

This amounts to proving that

(6.24)

\int_{\mathcal{M}_{T}}\mathscr{F}(\sqrt{-1}\partial\bar{\partial}\phi)\omega_{T}^{n}=0.

By Stokes’ theorem,

(6.25)

\int_{\mathcal{M}_{T}}(\omega_{T}+\sqrt{-1}\partial\bar{\partial}\phi)^{n}-\int_{\mathcal{M}_{T}}\omega_{T}^{n}=\int_{\partial\mathcal{M}_{T}}\gamma,

where $\gamma$ is the sum of terms involving one factor $d^{c}\phi$ and either $dd^{c}\phi$ or $\omega_{T}$ . We claim that $\gamma$ identically vanishes on $\partial\mathcal{M}_{T}$ . It suffices to show $\partial_{t}\lrcorner\gamma=0$ . Since by assumption $\phi$ is $S^{1}$ -invariant, so we have $\partial_{t}\phi=0$ . By the Neumann boundary condition, we also have $d^{c}\phi(\partial_{t})=0$ on $\partial\mathcal{M}_{T}$ . This follows from the observation that $J\partial_{t}=\nabla z$ . Now

(6.26)		$\displaystyle\partial_{t}\lrcorner\omega_{T}\|_{\partial\mathcal{M}_{T}}$	$\displaystyle=dz\|_{\partial\mathcal{M}_{T}}=0,$
(6.27)		$\displaystyle\partial_{t}\lrcorner dd^{c}\phi$	$\displaystyle=\mathcal{L}_{\partial_{t}}(d^{c}\phi)-d(\partial_{t}\lrcorner d^{c}\phi)=-d(\partial_{t}\lrcorner d^{c}\phi).$

The last term vanishes on $\partial\mathcal{M}_{T}$ since $d^{c}\phi(\partial_{t})=0$ pointwise on $\partial\mathcal{M}_{T}$ . ∎

Now we are ready to state the main result in this section.

Theorem 6.3 (Existence of $S^{1}$ -invariant Calabi-Yau metrics).

For each sufficiently large $T$ , there exists an $S^{1}$ -invariant Calabi-Yau metric $\omega_{T,CY}=\omega_{T}+\sqrt{-1}\partial\bar{\partial}\phi$ for $\phi\in\mathfrak{S}_{1}$ , such that

(6.28)

\|\sqrt{-1}\partial\bar{\partial}\phi\|_{\mathfrak{S}_{1}}\leq C_{0}\cdot T^{\nu+\alpha},

where $C_{0}>0$ is a uniform constant independent of $T\gg 1$ and the weighted Hölder norm of $\mathfrak{S}_{1}$ is defined in (6.8) for parameters $\nu$ , $\alpha$ , $\delta$ and $\mu$ satisfying (6.10), (6.11), (6.12) and (6.13).

To prove Theorem 6.3, we decompose the map $\mathscr{F}:\mathfrak{S}_{1}\to\mathfrak{S}_{2}$ as follows,

(6.29)

\mathscr{F}(\sqrt{-1}\partial\bar{\partial}\phi)-\mathscr{F}(\bm{0})=\mathscr{L}(\sqrt{-1}\partial\bar{\partial}\phi)+\mathscr{N}(\sqrt{-1}\partial\bar{\partial}\phi),

for any $\sqrt{-1}\partial\bar{\partial}\phi\in\mathfrak{S}_{1}$ , where

(6.30)		$\displaystyle\mathscr{L}(\sqrt{-1}\partial\bar{\partial}\phi)$	$\displaystyle=\Delta\phi,$
(6.31)		$\displaystyle\mathscr{N}(\sqrt{-1}\partial\bar{\partial}\phi)\cdot\omega_{T}^{n}$	$\displaystyle=(\omega_{T}+\sqrt{-1}\partial\bar{\partial}\phi)^{n}-\omega_{T}^{n}-n\omega_{T}^{n-1}\wedge\sqrt{-1}\partial\bar{\partial}\phi.$

By the definition of the weight function and Lemma 4.20, we have the following nonlinear error estimate.

Lemma 6.4 (Nonlinear error estimate).

For any sufficiently large $T\gg 1$ , let $\mathcal{M}_{T}$ be the neck endowed with the $C^{2,\alpha}$ -structure $(\omega_{T},\Omega_{T})$ . Then there exists a constant $C_{N}>0$ independent of $T\gg 1$ such that for all

(6.32)

\varrho\in(0,\frac{1}{2})

and

(6.33)

\sqrt{-1}\partial\bar{\partial}\phi_{2}\in\overline{B_{\varrho}(\bm{0})}\subset\mathfrak{S}_{1},\quad\sqrt{-1}\partial\bar{\partial}\phi_{2}\in\overline{B_{\varrho}(\bm{0})}\subset\mathfrak{S}_{1},

we have the pointwise estimate

(6.34)

\displaystyle\|\mathscr{N}(\sqrt{-1}\partial\bar{\partial}\phi_{1})-\mathscr{N}(\sqrt{-1}\partial\bar{\partial}\phi_{2})\|_{\mathfrak{S}_{2}}\leq C_{N}\cdot\varrho\cdot\|\sqrt{-1}\partial\bar{\partial}(\phi_{1}-\phi_{2})\|_{\mathfrak{S}_{1}}.

Proof.

By definition,

		$\displaystyle\Big(\mathscr{N}(\sqrt{-1}\partial\bar{\partial}\phi_{1})-\mathscr{N}(\sqrt{-1}\partial\bar{\partial}\phi_{2})\Big)\cdot\omega(t)^{n}$
(6.35)		$\displaystyle=$	$\displaystyle\sum\limits_{k=2}^{n}\begin{pmatrix}n\\ k\end{pmatrix}\cdot\omega(t)^{n-k}\wedge\Big((\sqrt{-1}\partial\bar{\partial}\phi_{1})^{k}-(\sqrt{-1}\partial\bar{\partial}\phi_{2})^{k}\Big).$

By the definition of the norm on $\mathfrak{S}_{1}$ , we have

(6.36)

\|\sqrt{-1}\partial\bar{\partial}\phi_{1}\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(\mathcal{M}_{T})}\leq\varrho,\quad\|\sqrt{-1}\partial\bar{\partial}\phi_{1}\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(\mathcal{M}_{T})}\leq\varrho.

With $\mu$ specified by (6.13), by Lemma 4.20, the weight function $\rho_{\delta,\nu+2,\mu}^{(\alpha)}:\mathcal{M}_{T}\to\mathbb{R}_{+}$ satisfies for any $\bm{x}\in\mathcal{M}_{T}$ ,

(6.37)

\rho_{\delta,\nu+2,\mu}^{(\alpha)}(\bm{x})\geq 1.

This implies the following weight-free estimates,

(6.38)

\|\sqrt{-1}\partial\bar{\partial}\phi_{1}\|_{C^{0,\alpha}(\mathcal{M}_{T})}\leq C_{0}\cdot\varrho\quad\text{and}\quad\|\sqrt{-1}\partial\bar{\partial}\phi_{2}\|_{C^{0,\alpha}(\mathcal{M}_{T})}\leq C_{0}\cdot\varrho,

where $C_{0}>0$ is a uniform constant independent of $T\gg 1$ .

Since the $L^{\infty}$ -norm of the Kähler form $\omega_{T}$ is bounded by a uniform constant (independent of $T\gg 1$ ), so the above estimates imply the pointwise estimate for $\mathscr{N}$ ,

(6.39)

|\mathscr{N}(\sqrt{-1}\partial\bar{\partial}\phi_{1})-\mathscr{N}(\sqrt{-1}\partial\bar{\partial}\phi_{2})|\leq C_{N}\cdot\varrho\cdot|\sqrt{-1}\partial\bar{\partial}(\phi_{1}-\phi_{2})|,

where $C_{N}>0$ is a uniform constant independent of $T$ . Write the above in terms of the weighted norms, we have

(6.40)

\|\mathscr{N}(\sqrt{-1}\partial\bar{\partial}\phi_{1})-\mathscr{N}(\sqrt{-1}\partial\bar{\partial}\phi_{2})\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(\mathcal{M}_{T})}\leq C_{N}\cdot\varrho\cdot\|\sqrt{-1}\partial\bar{\partial}(\phi_{1}-\phi_{2})\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(\mathcal{M}_{T})}.

The proof is done.

∎

To apply the implicit function theorem, we still need to prove the weighted linear estimate, which will be completed in the following subsections.

6.2. Some Liouville type theorems and removable singularity theorems

In this subsection, we introduce some removable singularity and Liouville type theorems, which will be needed in the proof of Proposition 7.15. For the convenience of discussions, we give precise statement here.

Lemma 6.5 (Removable singularity).

Let $(M^{n},g)$ be a Riemannian manifold such that $B_{R}(p)$ has a compact closure in $B_{2R}(p)$ . Let $K\subset M^{n}$ be a smooth submanifold with $\dim(K)=k_{0}\leq n-3$ . If $u$ is harmonic in $B_{R}(p)\setminus K$ and there is some $\epsilon\in(0,1)$ such that

(6.41)

|u(x)|\leq\frac{C}{d_{g}(x,K)^{(n-2-k_{0})-\epsilon}},

then $u$ is harmonic in $B_{R}(p)$ .

Proof.

The point is to apply integration by parts to show that $u$ is a weak solution to $\Delta u=0$ on $B_{R}(p)$ . The computations are routine and standard in the literature, so we just skip it. ∎

Lemma 6.6 (Liouville theorem on $\mathbb{R}^{m+n}$ ).

Given $m,n\in\mathbb{Z}_{+}$ with $m+n\geq 3$ , Let $\mu_{p}\in(-1,1)\setminus\{0\}$ and let $u\in C^{\infty}(\mathbb{R}^{m+n})$ be a harmonic function on the Euclidean space $(\mathbb{R}^{m+n},g_{\mathbb{R}^{m}}\oplus g_{\mathbb{R}^{n}})$ . If $u$ satsifies

(6.42)

\displaystyle|u(x,y)|\leq\frac{C}{|x|^{\mu_{p}}},\ \forall(x,y)\in(\mathbb{R}^{m}\setminus\{0\})\times\mathbb{R}^{n},

then $u\equiv 0$ on $\mathbb{R}^{m+n}$ .

Proof.

The proof is rather standard and straightforward, which can be achieved by using separation of variables.

For the simplicity of notations, we denote

(6.43)

d\equiv m+n\geq 3.

Let $(r,\Theta)\in\mathbb{R}^{d}$ be the polar coordinate system in $\mathbb{R}^{d}$ , so the Laplacian of $u$ can be written as

(6.44)

\Delta_{\mathbb{R}^{d}}u=\frac{\partial^{2}u}{\partial r^{2}}+\frac{d-1}{r}\cdot\frac{\partial u}{\partial r}+\frac{1}{r^{2}}\cdot\Delta_{\mathbb{S}^{d-1}}u.

We make separation of variables on the punctured Euclidean space $\mathbb{R}^{d}\setminus\{0^{d}\}$ . Let

(6.45)

\lambda_{j}\equiv j(j+d-2),\ j\in\mathbb{N},

be the spectrum of the unit round sphere $\mathbb{S}^{d-1}$ . Correspondingly, let $\varphi_{j}\in C^{\infty}(\mathbb{S}^{d-1})$ satisfy

(6.46)

-\Delta_{\mathbb{S}^{d-1}}\varphi_{j}(\Theta)=\lambda_{j}\varphi_{j}(\Theta).

Then the function $u(r,\Theta)$ has the expansion along the fiber $\mathbb{S}^{d-1}$ ,

(6.47)

u(r,\Theta)=\sum\limits_{j=0}^{\infty}u_{j}(r)\cdot\varphi_{j}(\Theta).

Immediately, for each $j\in\mathbb{N}$ , the coefficient function $u_{j}(r)$ solves the Euler-Cauchy equation,

(6.48)

u_{j}^{\prime\prime}(r)+\frac{d-1}{r}\cdot u_{j}^{\prime}(r)-\frac{1}{r^{2}}\cdot\lambda_{j}\cdot u_{j}(r)=0,

which has a general solution

(6.49)

u_{j}(r)=C_{j}\cdot r^{p_{j}}+C_{j}^{*}\cdot r^{q_{j}},

where $p_{j}=\frac{2-d+\sqrt{(d-2)^{2}+4\lambda_{j}}}{2}\geq 0$ and $q_{j}=\frac{2-d-\sqrt{(d-2)^{2}+4\lambda_{j}}}{2}<0$ solve the quadratic equation

(6.50)

w^{2}+(d-2)w-\lambda_{j}=0.

So it is obvious

	$\displaystyle p_{0}$	$\displaystyle=0,\quad q_{0}=2-d\leq-1,$
	$\displaystyle p_{j}$	$\displaystyle\geq p_{1}=1,$
(6.51)		$\displaystyle q_{j}$	$\displaystyle\leq q_{1}=1-d\leq-2,\quad j\in\mathbb{Z}_{+}.$

In the following, we will show that, given the growth condition (6.42) for $\mu_{p}\in(-1,1)\setminus\{0\}$ , then for each $j\in\mathbb{N}$ and for each $r>0$ , the coefficient $u_{j}(r)$ satisfies

(6.52)

|u_{j}(r)|\leq\frac{Q_{j}}{r^{\mu_{p}}},

where $Q_{j}\in\mathbb{R}$ . In fact, so it follows from the expansion (6.47) that for each $j\in\mathbb{N}$ ,

(6.53)

u_{j}(r)=\int_{\mathbb{S}^{d-1}}u(r,\Theta)\cdot\varphi_{j}\dvol_{\mathbb{S}^{d-1}},

which implies

(6.54)

|u_{j}(r)|\leq|\varphi_{j}|_{L^{\infty}(\mathbb{S}^{d-1})}\cdot\int_{\mathbb{S}^{d-1}}\frac{1}{|x|^{\mu_{p}}}\dvol_{\mathbb{S}^{d-1}}.

Next, we will write the above integral in the polar coordinates $\Theta\equiv(\theta_{1},\ldots,\theta_{d-1})$ with $\theta_{1},\ldots,\theta_{d-2}\in[0,\pi]$ and $\theta_{d-1}\in[0,2\pi]$ . Denote by $d\Theta\equiv d\theta_{1}\wedge d\theta_{2}\wedge\ldots\wedge d\theta_{d-1}$ , then it is by elementary calculations that, $|x|=r^{m}\cdot\prod\limits_{k=1}^{d-m}|\sin\theta_{k}|$ and $\dvol_{\mathbb{S}^{d-1}}=\prod\limits_{k=1}^{d-2}(\sin^{d-k-1}\theta_{k})\cdot d\Theta$ . Therefore,

(6.55)

\int_{\mathbb{S}^{d-1}}\frac{1}{|x|^{\mu_{p}}}\dvol_{\mathbb{S}^{d-1}}=\frac{1}{r^{\mu_{p}}}\int_{\mathcal{D}_{\Theta}}\frac{\prod\limits_{k=1}^{d-2}(\sin^{d-k-1}\theta_{k})}{\prod\limits_{k=1}^{d-m}|\sin\theta_{k}|^{\mu_{p}}}\cdot d\Theta,

where $\mathcal{D}_{\Theta}\equiv\{0\leq\theta_{1},\ldots,\theta_{d-2}\leq\pi,\ 0\leq\theta_{d-1}\leq 2\pi\}$ . By assumption, $\mu_{p}\in(-1,1)\setminus\{0\}$ , then $\frac{\prod\limits_{k=1}^{d-2}(\sin^{d-k-1}\theta_{k})}{\prod\limits_{k=1}^{d-m}|\sin\theta_{k}|^{\mu_{p}}}$ is integrable in $\mathcal{D}_{\Theta}$ and we denote

(6.56)

\mathcal{I}_{0}\equiv\int_{\mathcal{D}_{\Theta}}\frac{\prod\limits_{k=1}^{d-2}(\sin^{d-k-1}\theta_{k})}{\prod\limits_{k=1}^{d-m}|\sin\theta_{k}|^{\mu_{p}}}\cdot d\Theta.

Therefore, for each $j\in\mathbb{N}$ , it holds that

(6.57)

|u_{j}(r)|\leq\frac{\mathcal{I}_{0}\cdot|\varphi_{j}|_{L^{\infty}(\mathbb{S}^{d-1})}}{r^{\mu_{p}}}\equiv\frac{Q_{j}}{r^{\mu_{p}}}

for all $r>0$ .

Now we go back to the representation of $u_{j}(r)$ in (6.49) and we analyze the growth behavior of function as $r\ll 1$ and $r\gg 1$ . Applying the assumption $\mu_{p}\in(-1,1)\setminus\{0\}$ and the gap obtained in (6.51), we have that, for each $j\in\mathbb{N}$ , $C_{j}=C_{j}^{*}=0$ . Therefore,

(6.58)

u\equiv 0\ \text{on}\ \mathbb{R}^{m+n}.

∎

Lemma 6.7 (Liouville theorem on a cylinder).

Let $(Q,g_{Q})\equiv(D\times\mathbb{R},g_{Q})$ be a cylinder with a product Riemannian metric $g_{Q}=g_{D}\oplus dz^{2}$ , where $(D,g_{D})$ is a closed Riemannian manifold. Denote by $\lambda_{D}>0$ the lowest eigenvalue of the Laplace-Beltrami operator of $(D,g_{D})$ acting on functions. If $u$ is a harmonic function on $Q$ satisfying the growth control

(6.59)

|u|=O(e^{\lambda_{c}\cdot z})

for some $\lambda_{c}\in(0,\sqrt{\lambda_{D}})$ , then $u\equiv 0$ .

The proof follows from standard separation of variables, very similar to the proof of Proposition 3.31. We omit the details.

6.3. Weighted analysis and existence of incomplete Calabi-Yau metrics

We first prove

Proposition 6.8 (Uniform injectivity estimate on the neck).

For any sufficiently large parameter $T\gg 1$ , the linearized operator defined in (6.30)

(6.60)

\mathscr{L}:\mathfrak{S}_{1}\rightarrow\mathfrak{S}_{2},\quad\sqrt{-1}\partial\bar{\partial}\phi\mapsto\Delta\phi

is an isomorphism and satisfies the uniform injectivity estimate,

(6.61)

\|\sqrt{-1}\partial\bar{\partial}\phi\|_{\mathfrak{A}}\leq C_{L}\cdot\|\Delta\phi\|_{\mathfrak{B}}.

Here the constant $C_{L}>0$ is independent of the parameter $T\gg 1$ .

A preliminary ingredient in proving Proposition 6.8 is the following weighted Schauder estimate on $\mathcal{M}_{T}$ .

Proposition 6.9 (Weighted Schauder estimate on the neck, the global version).

For every sufficiently large parameter $T\gg 1$ , let $\mathcal{M}_{T}$ be the neck region with an $S^{1}$ -invariant Kähler metric $\omega_{T}$ constructed in Section 4.1. Then the following estimate hold:

(6.62)

\|u\|_{C_{\delta,\nu,\mu}^{2,\alpha}(\mathcal{M}_{T})}\leq C\Big(\|\Delta u\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(\mathcal{M}_{T})}+\Big\|\frac{\partial u}{\partial n}\Big\|_{C_{\delta,\nu+1,\mu}^{1,\alpha}}+\|u\|_{C_{\delta,\nu,\mu}^{0}(\mathcal{M}_{T})}\Big),

where the constant $C>0$ is independent of $T\gg 1$ .

Proof.

The proof follows directly from Proposition 4.22 and standard covering argument. We just skip the detailed proof.

∎

Next, the key part of the injectivity estimate in Proposition 6.8 is the following weighted estimate for higher derivatives with respect to the Neumann boundary value problem.

Proposition 6.10 (Uniform injectivity estimate on the neck).

Given a large parameter $T\gg 1$ , let $\mathcal{M}_{T}$ be the neck region with an $S^{1}$ -symmetric Kähler metric $\omega_{T}$ constructed in Section 4.1. Let the parameters $\mu$ , $\nu$ , $\alpha$ , $\delta$ satisfy satisfying

(6.63)

\displaystyle-1<\nu<0,\quad\nu+\alpha<0,\quad 0<\delta<\delta_{N},\ \mu=(1-\frac{1}{n})(\nu+2+\alpha)

as fixed in (6.10), (6.11), (6.12) and (6.13), then there exists a uniform constant $C>0$ (independent of $T$ ) such that for every $u\in C^{2,\alpha}(\mathcal{M}_{T})$ satisfying the boundary condition $\frac{\partial u}{\partial n}|_{\partial\mathcal{M}_{T}}=0$ , we have

(6.64)		$\displaystyle\\|\nabla u\\|_{C_{\delta,\nu+1,\mu}^{0}(\mathcal{M}_{T})}+\\|\nabla^{2}u\\|_{C_{\delta,\nu+2,\mu}^{0}(\mathcal{M}_{T})}\leq C\cdot\\|\Delta u\\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(\mathcal{M}_{T})},$
(6.65)		$\displaystyle[u]_{C_{\delta,\nu,\mu}^{2,\alpha}(\mathcal{M}_{T})}\leq C\cdot\\|\Delta u\\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(\mathcal{M}_{T})}.$

Proof.

The proof of the uniform estimate consists of two primary steps: In the first step, we will prove the weighted $C^{1}$ and $C^{2}$ estimates,

(6.66)

\|\nabla u\|_{C_{\delta,\nu+1,\mu}^{0}(\mathcal{M}_{T})}+\|\nabla^{2}u\|_{C_{\delta,\nu+2,\mu}^{0}(\mathcal{M}_{T})}\leq C\cdot\|\Delta u\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(\mathcal{M}_{T})}.

Next, based on the above weighted estimate and the weighted Schauder estimate (by Proposition 6.9), we will prove

(6.67)

\|\nabla u\|_{C_{\delta,\nu+1,\mu}^{0}(\mathcal{M}_{T})}+\|\nabla^{2}u\|_{C_{\delta,\nu+2,\mu}^{0}(\mathcal{M}_{T})}\leq C\cdot\|\Delta u\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(\mathcal{M}_{T})}.

Step 1. (Weighted $C^{1}$ and $C^{2}$ estimates)

Now we start to prove the estimate (6.66), which will be proved by contradiction. Suppose no such a uniform constant $C>0$ exists. That is, for fixed parameters

(6.68)

-1<\nu<0,\quad\nu+\alpha<0,\quad 0<\delta<\delta_{N},\quad\mu=(1-\frac{1}{n})(\nu+2+\alpha),

there are the following contradicting sequences:

(1)

A sequence of $S^{1}$ -invariant Kähler metrics $g_{j}=g_{T_{j}}$ (or $\omega_{j}=\omega_{T_{j}}$ ) on the neck $\mathcal{M}_{T_{j}}$ constructed in Section 4.1 with $T_{j}\to+\infty$ .

(2)

A sequence of $C^{2,\alpha}$ -functions $u_{j}\in\mathfrak{A}$ satisfying

(6.69)	$\displaystyle\frac{\partial u_{j}}{\partial n}\Big\|_{\mathcal{M}_{j}}$	$\displaystyle=0,$
(6.70)	$\displaystyle\\|\nabla u_{j}\\|_{C_{\delta,\nu+1,\mu}^{0}(\mathcal{M}_{j})}+\\|\nabla^{2}u_{j}\\|_{C_{\delta,\nu+2,\mu}^{0}(\mathcal{M}_{j})}$	$\displaystyle=1,$
(6.71)	$\displaystyle\\|\Delta u_{j}\\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(\mathcal{M}_{j})}$	$\displaystyle\to 0,\quad j\to+\infty.$

So it follows that either $\|\nabla u_{j}\|_{C_{\delta,\nu+1,\mu}^{0}(\mathcal{M}_{j})}\geq\frac{1}{2}$ or $\|\nabla^{2}u_{j}\|_{C_{\delta,\nu+2,\mu}^{0}(\mathcal{M}_{j})}\geq\frac{1}{2}$ . Without loss of generality, we only consider the first case and let $\bm{x}_{j}\in\mathcal{M}_{j}$ satisfy

(6.72)

|\rho_{\delta,\nu+1,\mu}^{(0)}(\bm{x}_{j})\cdot\nabla u_{j}(\bm{x}_{j})|=\|\nabla u_{j}\|_{C_{\delta,\nu+1,\mu}^{0}(\mathcal{M}_{j})}\geq\frac{1}{2}.

Now we renormalize the functions $u_{j}$ as follows,

(6.73)

v_{j}(\bm{x})=u_{j}(\bm{x})-u_{j}(\bm{x}_{j}).

Immediately, $v_{j}(\bm{x}_{j})=0$ , $\frac{\partial v_{j}}{\partial n}|_{\mathcal{M}_{j}}=0$ and

(6.74)	$\displaystyle\\|\nabla v_{j}\\|_{C_{\delta,\nu+1,\mu}^{0}(\mathcal{M}_{j})}$	$\displaystyle=\\|\nabla u_{j}\\|_{C_{\delta,\nu+1,\mu}^{0}(\mathcal{M}_{j})}\leq 1,$
(6.75)	$\displaystyle\\|\nabla^{2}v_{j}\\|_{C_{\delta,\nu+2,\mu}^{0}(\mathcal{M}_{j})}$	$\displaystyle=\\|\nabla^{2}u_{j}\\|_{C_{\delta,\nu+2,\mu}^{0}(\mathcal{M}_{j})}\leq 1,$
(6.76)	$\displaystyle\\|\Delta v_{j}\\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(\mathcal{M}_{j})}$	$\displaystyle=\\|\Delta u_{j}\\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(\mathcal{M}_{j})}\to 0,$
(6.77)	$\displaystyle\\|v_{j}\\|_{C_{\delta,\nu,\mu}^{0}(\mathcal{M}_{j})}$	$\displaystyle\leq C_{0}.$

So we are led to apply the weighted Schauder estimate in Proposition 6.9, which gives

(6.78)

\|v_{j}\|_{C_{\delta,\nu,\mu}^{2,\alpha}(\mathcal{M}_{j})}\leq C_{0}.

Moreover, it is straightforward that

(6.79)

\displaystyle|\rho_{\delta,\nu+1,\mu}^{(0)}(\bm{x}_{j})\cdot\nabla v_{j}(\bm{x}_{j})|=\|\nabla v_{j}\|_{C_{\delta,\nu+1,\mu}^{0}(\mathcal{M}_{j})}

\displaystyle\geq\frac{1}{2}.

We will rescale contradicting spaces $(\mathcal{M}_{j},g_{j})$ around the above reference points $\bm{x}_{j}$ such that the desired contradiction will arise in the limiting space. Let $g_{j}$ be a sequence of contradicting metrics, then we denote the rescaling factors as follows:

(1)

Rescaling of the metrics:

Let $\tilde{g}_{j}=\lambda_{j}^{2}\cdot g_{j}$ , then with respect to the fixed reference point $\bm{x}_{j}\in\mathcal{M}_{j}$ picked as the above, we have the convergence,

(6.80) $(\mathcal{M}_{j},\tilde{g}_{j},\bm{x}_{j})\xrightarrow{GH}(X_{\infty},\tilde{d}_{\infty},\bm{x}_{\infty}).$
(2)

Rescaling of the solutions:

Let $\kappa_{j}>0$ be a sequence of rescaling factors which will be determined later, such that

(6.81) $\tilde{v}_{j}\equiv\kappa_{j}\cdot v_{j}.$
(3)

Rescaling of the weight functions:

Denote by $\tilde{\rho}_{j,\delta,\nu,\mu}^{(k+\alpha)}$ and $\tilde{\rho}_{\infty,\delta,\nu,\mu}^{(k+\alpha)}$ the weight functions on the rescaled sequence $(\mathcal{M}_{j},\tilde{g}_{j},\bm{x}_{j})$ and the rescaled limit $(X_{\infty},\tilde{g}_{\infty},\bm{x}_{\infty})$ respectively. So we rescale the weight function $\rho_{j,\delta,\nu,\mu}^{(k+\alpha)}$ by

(6.82) $\tilde{\rho}_{j,\delta,\nu,\mu}^{(k+\alpha)}=\tau_{j}\cdot\rho_{j,\delta,\nu,\mu}^{(k+\alpha)}.$

Notice that the rescaling factor $\tau_{j}$ depends on $k$ and $\alpha$ .

In the following, we study the convergence of the renormalized functions $\tilde{v}_{j}\in\mathfrak{A}$ , with respect to the rescaled metrics $\tilde{g}_{j}$ , in each region according to the subdivision given in Section 4.3. The main goal is to show $\tilde{v}_{\infty}\equiv 0$ on the rescaled limit $X_{\infty}$ which gives the desired contradiction.

We will produce the desired contradiction in each region of $\bf{I}_{1}$ , $\bf{I}_{2}$ , $\bf{I}_{3}$ on $\mathcal{M}_{j}$ . Before the detailed contradiction arguments, let us determine the rescaling factors in the following way. First, the scaling invariance requires

(6.83)

\frac{\tau_{j}\cdot\kappa_{j}}{\lambda_{j}^{k+\alpha}}=1.

Now we need to combing the regularity scale analysis in Proposition 4.18 and the choice of the weight function in Definition 4.19. So $\lambda_{j}$ , $\tau_{j}$ and $\kappa_{j}$ are determined as follows, which depends on if $|z(\bm{x}_{j})|$ is uniformly bounded: First, if $|z(\bm{x}_{j})|$ is uniformly bounded (corresponding to Region $\bf{I}_{1}$ , $\bf{I}_{2}$ and Case (a) of Region $\bf{I}_{3}$ ), we choose

(6.84)

\displaystyle\begin{cases}\lambda_{j}=\mathfrak{s}_{j}^{-1}\\ \tau_{j}=(\mathfrak{s}_{j}^{-1})^{\nu+k+\alpha}\cdot T_{j}^{-\mu}\\ \kappa_{j}=(\mathfrak{s}_{j}^{-1})^{-\nu}\cdot T_{j}^{\mu}.\end{cases}

Next, if $|z(\bm{x}_{j})|\to+\infty$ (corresponding to Case (b) and Case (c) of Region $\bf{I}_{3}$ ), we choose

(6.85)

\displaystyle\begin{cases}\lambda_{j}=\mathfrak{s}_{j}^{-1}\\ \tau_{j}=(\mathfrak{s}_{j}^{-1})^{\nu+k+\alpha}\cdot e^{-T_{j}}\cdot T_{j}^{-\mu}\\ \kappa_{j}=(\mathfrak{s}_{j}^{-1})^{-\nu}\cdot e^{T_{j}}\cdot T_{j}^{\mu}.\end{cases}

In this case, we need to rescale the $z$ -coordinate in the meanwhile so that the exponential term shows up in the rescaling factors.

Region $\bf{I}_{1}$ (The deepest bubble):

In this case, we consider that the reference points $\bm{x}_{j}$ are in Region $\bf{I}_{1}$ . According to the discussions in Section 4.3, for any $0<\gamma<1$ , $(\mathcal{M}_{j},\tilde{g}_{j},\bm{x}_{j})$ converges to the following Riemann product in the $C^{2,\gamma}$ -topology,

(6.86)

(\mathcal{M}_{j},\tilde{g}_{j},\bm{x}_{j})\xrightarrow{C^{2,\gamma}}(\mathbb{C}_{TN}^{2}\times\mathbb{C}^{n-2},\tilde{g}_{\infty},\bm{x}_{\infty}),

where $\tilde{g}_{\infty}\equiv g_{TN}\oplus g_{\mathbb{C}^{n-2}}$ is the product metric of the Taub-NUT metric $g_{TN}$ and the Euclidean metric $g_{\mathbb{C}^{n-2}}$ . Moreover, the rescaled weight function will converge to

(6.87)

\displaystyle\tilde{\rho}_{\infty,\delta,\nu,\mu}^{(k+\alpha)}(\bm{x})=\begin{cases}1,&\bm{x}\in T_{1}(\Sigma_{0}),\\ (d_{\tilde{g}_{\infty}}(\bm{x},\Sigma_{0}))^{\nu+k+\alpha},&\bm{x}\in(\mathbb{C}_{TN}^{2}\times\mathbb{C}^{n-2})\setminus T_{1}(\Sigma_{0}),\end{cases}

where $\Sigma_{0}\equiv\{p_{\infty}\}\times\mathbb{C}^{n-2}\subset\mathbb{C}_{TN}^{2}\times\mathbb{C}^{n-2}$ for some $p_{\infty}\in\mathbb{C}_{TN}^{2}$ , is the Gromov-Hausdorff limit of the lifted divisor $\mathcal{P}\equiv\pi^{-1}(P)\subset\mathcal{M}_{j}$ with respect to the rescaled metrics $\tilde{g}_{j}$ such that and

(6.88)

T_{1}(\Sigma_{0})\equiv\{\bm{x}\in\mathbb{C}_{TN}^{2}\times\mathbb{C}^{n-2}|d_{\tilde{g}_{\infty}}(\bm{x},\Sigma_{0})\leq 1\}.

It is straightforward that, the rescaled functions $\tilde{v}_{j}$ converge to $\tilde{v}_{\infty}$ in the $C^{2,\alpha^{\prime}}$ -topology for each $0<\alpha^{\prime}<\alpha$ such that the following properties hold,

(1)

$\|\nabla\tilde{v}_{\infty}\|_{C_{\delta,\nu+1,\mu}^{0}(\mathbb{C}_{TN}^{2}\times\mathbb{C}^{n-2},\tilde{g}_{\infty})}+\|\nabla^{2}\tilde{v}_{\infty}\|_{C_{\delta,\nu+2,\mu}^{0}(\mathbb{C}_{TN}^{2}\times\mathbb{C}^{n-2},\tilde{g}_{\infty})}=1$ ,
(2)

$\tilde{v}_{\infty}(\bm{x}_{\infty})=0$ ,
(3)

$\Delta_{\tilde{g}_{\infty}}\tilde{v}_{\infty}\equiv 0$ on $\mathbb{C}_{TN}^{2}\times\mathbb{C}^{n-2}$ .

We will prove that $\tilde{v}_{\infty}\equiv 0$ on $\mathbb{C}_{TN}^{2}\times\mathbb{C}^{n-2}$ .

To start with, we will show that $\tilde{v}_{\infty}$ is constant on the Euclidean factor $\mathbb{C}^{n-2}$ . Indeed, we write $\bm{x}\equiv(\bm{x}^{\prime},\bm{x}^{\prime\prime})\in\mathbb{C}_{TN}^{2}\times\mathbb{C}^{n-2}$ , so it suffices to prove that for every $1\leq k\leq 2n-4$ , we have

(6.89)

|\nabla_{k}\tilde{v}_{\infty}|\equiv 0\ \text{on}\ \mathbb{C}^{n-2},

where the partial derivative $\nabla_{k}\tilde{v}_{\infty}(\bm{x})\equiv\frac{\partial\tilde{v}_{\infty}}{\partial x_{k}^{\prime\prime}}(\bm{x}^{\prime},\bm{x}^{\prime\prime})$ is taken in the directions of $\mathbb{C}^{n-2}$ . Now for every $1\leq k\leq 2n-4$ ,

(6.90)

\Delta_{\tilde{g}_{\infty}}(\nabla_{k}\tilde{v}_{\infty})=\Delta_{\mathbb{C}_{TN}^{2}}(\nabla_{k}\tilde{v}_{\infty})+\Delta_{\mathbb{C}^{n-2}}(\nabla_{k}\tilde{v}_{\infty}).

Notice that $\tilde{g}_{\infty}=g_{TN}\oplus g_{\mathbb{C}^{n-2}}$ is a product metric and $\nabla_{k}$ in effect acts on the Euclidean factor $\mathbb{C}^{n-2}$ , so $\nabla_{k}$ commutes with both $\Delta_{\mathbb{C}_{TN}^{2}}$ and $\Delta_{\mathbb{C}^{n-2}}$ . Therefore,

(6.91)

\displaystyle\Delta_{\tilde{g}_{\infty}}(\nabla_{k}\tilde{v}_{\infty})=\nabla_{k}(\Delta_{\mathbb{C}_{TN}^{2}}\tilde{v}_{\infty}+\Delta_{\mathbb{C}^{n-2}}\tilde{v}_{\infty})=0.

The weighted bound implies the estimates

(6.92)

\displaystyle\begin{cases}|\nabla_{k}\tilde{v}_{\infty}(\bm{x})|\leq 1,&d_{\tilde{g}_{\infty}}(\bm{x},\Sigma_{0})\leq 1,\\ |\nabla_{k}\tilde{v}_{\infty}(\bm{x})|\leq d_{\tilde{g}_{\infty}}(\bm{x},\Sigma_{0})^{-(\nu+1)},&d_{\tilde{g}_{\infty}}(\bm{x},\Sigma_{0})\geq 1.\end{cases}

Since we have assumed $\nu\in(-1,0)$ , so it is straightforward

(6.93)

-(\nu+1)\in(-1,0).

The above implies that $|\nabla_{k}\tilde{v}_{\infty}|\leq 1$ on $\mathbb{C}_{TN}^{2}\times\mathbb{C}^{n-2}$ . Applying Cheng-Yau’s gradient estimate to the harmonic function $\nabla_{k}\tilde{v}_{\infty}$ on the Ricci-flat manifold $\mathbb{C}_{TN}^{2}\times\mathbb{C}^{n-2}$ , we conclude that $\nabla_{k}\tilde{v}_{\infty}$ is constant on $\mathbb{C}_{TN}^{2}\times\mathbb{C}^{n-2}$ . By (6.92), $\nabla_{k}\tilde{v}_{\infty}\equiv 0$ for every $1\leq k\leq 2n-4$ . Therefore, $\tilde{v}_{\infty}$ is constant on the Euclidean factor $\mathbb{C}^{n-2}$ .

By the above argument, the limiting function $\tilde{v}_{\infty}$ can be viewed as a harmonic function on the Ricci-flat Taub-NUT space $(\mathbb{C}_{TN}^{2},g_{TN})$ . Now applying Bochner’s formula,

(6.94)

\frac{1}{2}\Delta_{g_{TN}}|\nabla_{g_{TN}}\tilde{v}_{\infty}|^{2}=|\nabla_{g_{TN}}^{2}\tilde{v}_{\infty}|^{2}\geq 0.

Since $\tilde{v}_{\infty}$ satisfies the weighted bound

(6.95)

\|\nabla_{g_{TN}}\tilde{v}_{\infty}\|_{C_{\delta,\nu+1,\mu}^{0}(\mathbb{C}_{TN}^{2})}+\|\nabla_{g_{TN}}^{2}\tilde{v}_{\infty}\|_{C_{\delta,\nu+2,\mu}^{0}(\mathbb{C}_{TN}^{2})}=1,

so we have for any $\bm{x}\in\mathbb{C}_{TN}^{2}\setminus B_{1}(\bm{x}_{\infty})$ ,

(6.96)

|\nabla_{g_{TN}}\tilde{v}_{\infty}(\bm{x})|\leq d_{g_{TN}}(\bm{x},\bm{x}_{\infty})^{-(\nu+1)}.

By assumption $\nu\in(-1,0)$ , then $|\nabla_{g_{TN}}\tilde{v}_{\infty}|\equiv 0$ on $\mathbb{C}_{TN}^{2}$ and hence $\tilde{v}_{\infty}$ is constant on $\mathbb{C}_{TN}^{2}$ . Notice that $\tilde{v}_{\infty}(\bm{x}_{\infty})=0$ , so we conclude that $\tilde{v}_{\infty}(\bm{x}_{\infty})\equiv 0$ .

Region $\bf{I}_{2}$ (bubble transformations):

Now we separate the proof in $3$ cases:

(a)

There is some $\sigma_{0}>0$ such that

(6.97) $\lambda_{j}\cdot T_{j}^{-\frac{1}{2}}\geq\sigma_{0}.$
(b)

Assume that $r_{j}$ satisfies the following condition holds,

(6.98) $\lambda_{j}\cdot T_{j}^{-\frac{1}{2}}\to 0,\ \lambda_{j}\cdot T_{j}^{\frac{1}{2}}\to\infty.$
(c)

Assume that there is some $C_{0}>0$ such that $\lambda_{j}\cdot T_{j}^{\frac{1}{2}}\leq C_{0}$ .

Case (a):

In this case, the rescaled limit is the Riemann product $\mathbb{C}_{TN,\sigma}^{2}\times\mathbb{C}^{n-2}$ , where $\mathbb{C}_{TN,\sigma}^{2}$ is the Taub-NUT space and the length of the circle fiber at infinity equals $\sigma\in[\sigma_{0},1]$ . The remainder of the proof is the same as that in Region $\bf{I}_{1}$ , so we omit it.

Case (b):

In this case, the rescaled spaces $(\mathcal{M}_{j},\tilde{g}_{j},\bm{x}_{j})$ converge to the product Euclidean space $(\mathbb{R}^{3}\times\mathbb{C}^{n-2},g_{0},\bm{x}_{\infty})$ in the pointed Gromov-Hausdorff topology, i.e.,

(6.99)

(\mathcal{M}_{j},\tilde{g}_{j},\bm{x}_{j})\xrightarrow{GH}(\mathbb{R}^{3}\times\mathbb{C}^{n-2},g_{0},\bm{x}_{\infty}),

where the metric $g_{0}$ is the standard Euclidean metric on $\mathbb{R}^{3}\times\mathbb{C}^{n-2}$ . In this rescaled limit, the limiting reference point $\bm{x}_{\infty}$ satisfies $d_{g_{0}}(\bm{x}_{\infty},\Sigma_{0^{3}})=1$ and $\Sigma_{0^{3}}\equiv\{0^{3}\}\times\mathbb{C}^{n-2}\subset\mathbb{R}^{3}\times\mathbb{C}^{n-2}$ is the singular slice. Moreover, the convergence keeps curvatures uniformly bounded away from the singular slice $\Sigma_{0^{3}}$ . By passing to the local universal covers, in fact one can show that, away from $\Sigma_{0^{3}}\subset\mathbb{R}^{3}\times\mathbb{C}^{n-2}$ , the rescaled contradicting functions $\tilde{v}_{j}$ converge to $\tilde{v}_{\infty}$ in the $C^{2,\alpha^{\prime}}$ -topology for each $0<\alpha^{\prime}<\alpha<1$ , such that the following properties hold,

(1)

$\|\nabla\tilde{v}_{\infty}\|_{C_{\delta,\nu+1,\mu}^{0}(\mathbb{R}^{3}\times\mathbb{C}^{n-2})}+\|\nabla^{2}\tilde{v}_{\infty}\|_{C_{\delta,\nu+2,\mu}^{0}(\mathbb{R}^{3}\times\mathbb{C}^{n-2})}=1$ ,
(2)

$\tilde{v}_{\infty}(\bm{x}_{\infty})=0$ ,
(3)

$\Delta_{\tilde{g}_{\infty}}\tilde{v}_{\infty}\equiv 0$ in $(\mathbb{R}^{3}\times\mathbb{C}^{n-2})\setminus\Sigma_{0^{3}}$ ,

where the limiting weight function is

(6.100)

\rho_{\infty,\delta,\nu,\mu}^{(k+\alpha)}(\bm{x})=(d_{g_{0}}(\bm{x},\Sigma_{0^{3}}))^{\nu+k+\alpha},\ \bm{x}\in\mathbb{R}^{3}\times\mathbb{C}^{n-2}.

Our goal is to show that $\tilde{v}_{\infty}\equiv 0$ on $\mathbb{R}^{3}\times\mathbb{C}^{n-2}$ , which consists of the following ingredients:

First, we will prove that $\tilde{v}_{\infty}$ in fact globally harmonic in $\mathbb{R}^{3}\times\mathbb{C}^{n-2}$ . To show the singular slice $\Sigma_{0^{3}}$ is removable, for each $q\in\Sigma_{0^{3}}$ , we take a unit ball $B_{1}(q)\subset\Sigma_{0^{3}}$ , and for any $r\in(0,1)$ , we choose the tubular neighborhood $T_{r}(B_{1}(q))\subset\mathbb{R}^{3}\times\mathbb{C}^{n-2}$ . Notice that $\nabla\tilde{v}_{\infty}$ satisfies the uniform estimate

(6.101)

\|\nabla\tilde{v}_{\infty}\|_{C_{\delta,\nu+1,\mu}^{0}(\mathbb{R}^{3}\times\mathbb{C}^{n-2})}\leq 1,

integrating the above weighted bound, then for any $\bm{x}\in T_{r}(B_{1}(q))\setminus B_{1}(q)$ ,

(6.102)

|\tilde{v}_{\infty}(\bm{x})|\leq C\cdot d(\bm{x},B_{1}(q))^{-(\nu)}.

By Lemma 6.5, $B_{1}(q)$ is a removable singular set in $T_{r}(B_{1}(q))$ and hence $\tilde{v}_{\infty}$ is harmonic in $T_{r}(B_{1}(q))$ .

Next, we will show that $\tilde{v}_{\infty}$ is constant in $\mathbb{C}^{n-2}$ . It is straightforward that for each $1\leq k\leq 2n-4$ , the partial derivative $\nabla_{k}\tilde{v}_{\infty}\equiv\frac{\partial}{\partial x_{k}^{\prime\prime}}\tilde{v}_{\infty}(\bm{x}^{\prime},\bm{x}^{\prime\prime})$ satisfies

(6.103)

\Delta_{g_{0}}(\nabla_{k}\tilde{v}_{\infty})=0\ \text{in}\ \mathbb{R}^{3}\times\mathbb{C}^{n-2}.

The weighted condition implies that $\nabla_{k}\tilde{v}_{\infty}$ satisfies the uniform estimate,

(6.104)

|\nabla_{k}\tilde{v}_{\infty}|\leq d(\bm{x},\Sigma_{0^{3}})^{-(\nu+1)},\forall\bm{x}\in\mathbb{R}^{3}\times\mathbb{C}^{n-2}.

Since we have assumed $\nu\in(-1,0)$ , Lemma 6.6 implies that $|\nabla_{k}\tilde{v}_{\infty}|\equiv 0$ on $\mathbb{R}^{3}\times\mathbb{C}^{n-2}$ and hence $\tilde{v}_{\infty}$ is constant in $\mathbb{C}^{n-2}$ . Therefore, $\tilde{v}_{\infty}$ can be viewed as a harmonic function in the Euclidean space $(\mathbb{R}^{3},g_{\mathbb{R}^{3}})$ . By assumption, $\tilde{v}_{\infty}$ satisfies

(6.105)

|\tilde{v}_{\infty}(\bm{x})|\leq d_{g_{\mathbb{R}^{3}}}(\bm{x},0^{3})^{-\nu}.

Since $\nu\in(-1,0)$ , applying the standard Liouville theorem for sublinear growth harmonic functions on a Euclidean space, we conclude that $\tilde{v}_{\infty}$ is a constant. The last step is to use the renormalization $\tilde{v}_{\infty}(\bm{x}_{\infty})=0$ , then $\tilde{v}_{\infty}\equiv 0$ .

Case (c):

The rescaled limit is the cylinder $(Q,g_{c})\equiv(D\times\mathbb{R},g_{D}\oplus dz^{2})$ , where $(D,g_{D})$ is a closed Calabi-Yau manifold. The limiting solutions $\tilde{v}_{\infty}$ satisfies

(1)

$\|\nabla\tilde{v}_{\infty}\|_{C_{\delta,\nu+1,\mu}^{0}(Q)}+\|\nabla^{2}\tilde{v}_{\infty}\|_{C_{\delta,\nu+2,\mu}^{0}(Q)}=1$ ,
(2)

$\tilde{v}_{\infty}(\bm{x}_{\infty})=0$ ,
(3)

$\Delta_{\tilde{g}_{\infty}}\tilde{v}_{\infty}\equiv 0$ in $Q\setminus P$ ,

where the limiting weight function is

(6.106)

\displaystyle\tilde{\rho}_{\infty,\delta,\nu,\mu}^{(k+\alpha)}(\bm{x})=\begin{cases}e^{\delta\cdot z(\bm{x})}\cdot\mathfrak{r}(\bm{x})^{\nu+k+\alpha},&z(\bm{x})>0\\ e^{-\delta\cdot z(\bm{x})}\cdot\mathfrak{r}(\bm{x})^{\nu+k+\alpha},&z(\bm{x})\leq 0.\end{cases}

Similar to Case (b), first we need to extend the limiting function $\tilde{v}_{\infty}$ across the singular set $P$ . Integrating $\nabla\tilde{v}_{\infty}$ around $P$ , we have that $\tilde{v}_{\infty}$ satisfies the growth estimate

(6.107)

|\tilde{v}_{\infty}(\bm{x})|\leq C\cdot d_{g_{c}}(\bm{x},P)^{-\nu}.

Since we have assumed $\nu\in(-1,0)$ , so Lemma 6.5 implies that the singular set $P$ is removable. Now we have obtained that $\tilde{v}_{\infty}$ is harmonic on $Q$ and satisfies

(6.108)

|\tilde{v}_{\infty}(\bm{x})|\leq Ce^{-\delta\cdot|z(\bm{x})|},

for $|z(\bm{x})|$ large. Therefore, $\tilde{v}_{\infty}\equiv 0$ on $Q$ which completes the proof of Case (c).

Region $\bf{I}_{3}$ (the cylindrical bubble and the boundary behavior):

In this region, the rescaling factors of the metrics $g_{j}$ are chosen such that the rescaled Gromov-Hausdorff limit is the cylinder $Q\equiv D\times\mathbb{R}$ . Let $\zeta_{j}\equiv z(\bm{x}_{j})$ , then there are two different cases to analyze which depends on if the convergence keeps curvatures uniformly bounded.

(a)

Assume that there is some $\zeta_{0}>0$ such that $|z_{j}|\leq\zeta_{0}$ .
(b)

Assume that $z_{j}$ satisfies

(6.109) $|\zeta_{j}|\to\infty,\ \frac{T_{j}^{\frac{n-2}{n}}}{L_{T_{j}}(z_{j})}\to 0.$
(c)

Assume that $z_{j}$ satisfies

(6.110) $\displaystyle c_{0}\leq\frac{T_{j}^{\frac{n-2}{n}}}{L_{T_{j}}(z_{j})}\leq 1.$

Case (a):

So the rescaled spaces $(\mathcal{M}_{j},\tilde{g}_{j},\bm{x}_{j})$ converge to the cylinder $(Q,g_{c})=(D^{2n}\times\mathbb{R},g_{D^{2n}}\oplus dz^{2})$ and the sequence has uniformly bounded geometry away from $Q\setminus\mathcal{P}$ . Moreover, the weight function in the rescaled limit space is

(6.111)

\displaystyle\tilde{\rho}_{\infty,\delta,\nu,\mu}^{(k+\alpha)}(\bm{x})=\begin{cases}e^{\delta\cdot z(\bm{x})}\cdot\mathfrak{r}(\bm{x})^{\nu+k+\alpha},&z(\bm{x})>0\\ e^{-\delta\cdot z(\bm{x})}\cdot\mathfrak{r}(\bm{x})^{\nu+k+\alpha},&z(\bm{x})\leq 0.\end{cases}

The rest of the proof is the same as Case (c) in Region II.

Case (b) in Region $\bf{I}_{3}$

In this case, the reference point $\bm{x}_{j}$ satisfies

(6.112)

|z(\bm{x}_{j})|\to\infty,\quad\frac{T_{j}^{\frac{n-2}{n}}}{L_{T_{j}}(z_{j})}\to 0.

In addition, we also need to perform the coordinate change centered at the reference point $\bm{x}_{j}$ ,

(6.113)

z(\bm{x})=z_{j}+\Big(\frac{T_{j}}{L_{T_{j}}(z_{j})}\Big)^{\frac{n-2}{2}}w(\bm{x}).

In the following, we only consider the case $z_{j}\ll 0$ . It is shown in Section 4.3 that the rescaled limit is isometric to a cylinder $Q=D\times\mathbb{R}$ with a product metric

(6.114)

g_{Q}=g_{D}+dw^{2}.

Moreover, as $T_{j}\to+\infty$ , the rescaled weight function limits to

(6.115)

\tilde{\rho}_{\infty,\delta,\nu,\mu}^{(k+\alpha)}(\bm{x})=e^{-\frac{\delta\cdot n\cdot k_{-}}{2}\cdot w(\bm{x})}.

Now the growth condition implies that the limiting function $\tilde{v}_{\infty}$ satisfies

(6.116)

\displaystyle\begin{cases}\Delta_{Q}\tilde{v}_{\infty}(\bm{x})=0,&\forall\bm{x}\in Q,\\ |\tilde{v}_{\infty}(\bm{x})|\leq e^{\frac{\delta\cdot n\cdot k_{-}}{2}\cdot w(\bm{x})},&w\in\mathbb{R}.\end{cases}

By the choice of the parameter $\delta$ in (6.12),

(6.117)

\frac{\delta\cdot n\cdot k_{-}}{2}<\frac{\sqrt{\lambda_{D}}}{2}.

Applying Lemma 6.7, for every $\bm{x}\in Q$ ,

(6.118)

\tilde{v}_{\infty}(\bm{x})=0.

So the proof of Case (b) is done.

Case (c) in Region $\bf{I}_{3}$

In this case, the reference point $\bm{x}_{j}$ is close to the boundary such that Neumann boundary condition plays a crucial role. Precisely, the scale condition is given by the following: there is some $c_{0}>0$ such that

(6.119)

\displaystyle c_{0}\leq\frac{T_{j}^{\frac{n-2}{n}}}{L_{T_{j}}(z_{j})}\leq 1.

We can assume that $z_{j}\ll 0$ and passing to a subsequence, there is some constant $\mathfrak{c}_{0}\in[c_{0},1]$ such that

(6.120)

\frac{T_{j}^{\frac{n-2}{n}}}{L_{T_{j}}(z_{j})}\to\mathfrak{c}_{0}.

For the convenience of the computations, we will perform the coordinate change centered at the boundary slice, that is,

(6.121)

z(\bm{x})=T_{-}+\Big(\frac{T_{j}}{L_{T_{j}}(z_{j})}\Big)^{\frac{n-2}{2}}w(\bm{x}).

We have computed in Section 4.3 that the limit of the rescaled spaces $(\mathcal{M}_{T},\tilde{g}_{j},\bm{x}_{j})$ is the Calabi model space $(\mathcal{C}^{n}_{-},g_{\mathcal{C}^{n}_{-}},\bm{x}_{\infty})$ . Moreover, the limiting weight function is

(6.122)

\tilde{\rho}_{\infty,\delta,\nu,\mu}^{(k+\alpha)}(\bm{x})=e^{-\delta\cdot\mathfrak{c}_{0}^{\frac{n}{2}}\cdot P_{{\frac{n}{2}}}(w)}\cdot(P_{\frac{n}{2}}(w))^{\frac{\nu+k+\alpha}{2}},

where

(6.123)

P_{{\frac{n}{2}}}(w)\equiv(1+k_{-}\mathfrak{c}_{0}^{\frac{n}{2}}w)^{\frac{n}{2}}.

Since $\mathfrak{c}_{0}\in[c_{0},1]$ , so the limiting function $\tilde{v}_{\infty}$ satisfies

(6.124)

\displaystyle\begin{cases}\Delta_{g_{\mathcal{C}_{-}^{n}}}\tilde{v}_{\infty}=0,&\bm{x}\in\mathcal{C}_{-}^{n},\\ |\tilde{v}_{\infty}(\bm{x})|\leq e^{\delta\cdot(k_{-})^{\frac{n}{2}}\cdot w^{\frac{n}{2}}}\cdot P_{\frac{n}{2}}(w)^{-\frac{\nu}{2}},&w(\bm{x})\gg 1,\\ \frac{\partial\tilde{v}_{\infty}}{\partial w}=0,&w(\bm{x})=w_{0}.\end{cases}

In the following, we will prove that $\tilde{v}_{\infty}$ is vanishing everywhere in the Calabi space $\mathcal{C}_{-}^{n}$ such that the contradiction arises.

To see this, recall that the incomplete Calabi model space $(\mathcal{C}_{-}^{n},g_{\mathcal{C}_{-}})$ is diffeomorphic to the topological product $[w_{0},+\infty)\times Y^{2n-1}$ , where $Y^{2n-1}\equiv\{\rho=\rho_{0}\}$ is with respect to the boundary slice in the Calabi model (see Section 5 for detailed discussions on it). The above structure leads to a natural coordinate representation $\bm{x}=(w,\bm{y})\in\mathcal{C}_{-}^{n}$ for each point in the Calabi model space such that the boundary of $\mathcal{C}_{-}^{n}$ is given by $\{w=w_{0}\}$ , where the coordinate $w$ is the natural moment map coordinate.

Denote by $\Sigma_{Y^{2n-1}}=\{\Lambda_{k}\}_{k=0}^{\infty}$ the spectrum of the fiber $Y^{2n-1}$ with respect to the induced Riemannian metric. Let $\{\varphi_{k}\}_{k=0}^{\infty}$ be the orthonormal basis with respect to the $L^{2}$ -inner product on $Y^{2n-1}$ , such that for each $k\in\mathbb{N}$ ,

(6.125)

\displaystyle-\Delta_{Y^{2n-1}}\varphi_{k}=\lambda_{k}\cdot\varphi_{k}.

If $\delta>0$ is chosen sufficiently small, applying Proposition 5.14, then $\tilde{v}_{\infty}$ has the expansion

(6.126)

\tilde{v}_{\infty}(w,\bm{y})=\kappa\cdot w+\ell+\sum\limits_{k=1}^{\infty}c_{k}\cdot\mathcal{D}_{k}(w)\cdot\varphi_{k}(\bm{y}),

where the function $\mathcal{D}_{k}(w)$ has some definite exponential decaying rate (see Lemma 5.4 and Lemma 5.7 for the accurate rates).

Now we apply the Neumann condition to show that $\kappa=0$ and $c_{k}=0$ for all $k\in\mathbb{N}$ . In fact,

(6.127)

\frac{\partial\tilde{v}_{\infty}(w,\bm{y})}{\partial w}=\kappa+\sum\limits_{k=1}^{\infty}c_{k}\cdot\mathcal{D}_{k}^{\prime}(w)\cdot\varphi_{k}(\bm{y}).

Integrating (6.127) over the boundary slice $\{w=w_{0}\}$ ,

(6.128)

\kappa\cdot\Vol_{g_{\mathcal{C}_{-}^{n}}}(Y^{2n-1})=\int_{Y^{2n-1}}\frac{\partial\tilde{v}_{\infty}(w,\bm{y})}{\partial w}\Big|_{w=w_{0}}=0,

which implies

(6.129)

\kappa=0.

Next, for each fixed $k\in\mathbb{Z}_{+}$ , multiplying $\varphi_{k}$ on the both sides of (6.127) and integrating over $Y^{2n-1}$ ,

(6.130)

c_{k}\cdot\mathcal{D}_{k}^{\prime}(w)=\int_{Y^{2n-1}}\varphi_{k}(\bm{y})\cdot\frac{\partial\tilde{v}_{\infty}(w,\bm{y})}{\partial w}\Big|_{w=w_{0}}=0.

The conclusion $c_{k}=0$ follows from the claim

(6.131)

\mathcal{D}_{k}^{\prime}(w)<0,\forall\ w\geq w_{0}.

Now we just need to prove the claim. In fact, since $\mathcal{D}_{k}^{\prime}(w)$ satisfies the equation

(6.132)

\mathcal{D}_{k}^{\prime\prime}(w)=w^{n-2}(\frac{j_{k}^{2}n^{2}}{4}\cdot w^{n}+n\lambda_{k})\mathcal{D}_{k}(w)

and hence

(6.133)

\mathcal{D}_{k}^{\prime\prime}(w)>0.

Notice that $\mathcal{D}_{k}(w)$ has an exponential decaying rate. This tells us that $\mathcal{D}_{k}^{\prime\prime}(w)>0$ and bounded as $w\to+\infty$ . Therefore, $\mathcal{D}_{k}^{\prime}(w)$ is increasing and uniformly continuous for $w>0$ . Since $\mathcal{D}_{k}(w)\to 0$ , we conclude that $\lim\limits_{w\to+\infty}\mathcal{D}_{k}^{\prime}(w)=0$ . Therefore, $\mathcal{D}_{k}(w)<0$ for any $w\geq w_{0}$ .

Lastly, $\tilde{v}_{\infty}$ satisfies the renormalization condition $\tilde{v}_{\infty}(\bm{x}_{\infty})=0$ , immediately, $\ell=0$ . Therefore,

(6.134)

\tilde{v}_{\infty}\equiv 0\quad\text{on}\ \mathcal{C}_{-}^{n}.

The proof is done.

∎

Combining all the above estimates, we are ready to complete the proof of Theorem 6.3.

Proof of Theorem 6.3.

It suffices to verify each condition for $\mathscr{F}$ in Lemma 6.1. Proposition 6.8 and Proposition 6.4 show that $\mathscr{F}$ satisfies Item (1) and Item (2a). In our context, $C_{L}>0$ and $C_{N}>0$ are uniform constants. $r_{0}>0$ can be chosen as any fixed constant in $(0,\frac{1}{2C_{L}C_{N}})$ . To verify Item (2b) in Lemma 6.1, we just need to use (6.21). In fact, we have assumed $\nu+\alpha<0$ , then

(6.135)

\|\mathscr{F}(\bm{0})\|_{\mathfrak{S}_{2}}\leq C\cdot T^{\nu+\alpha}\ll\frac{r_{0}}{4C_{L}},

as $T$ is sufficiently large. This completes the proof.

∎

6.4. Geometric singularity and normalized limit measure

The goal of this subsection is to understand the measured Gromov-Hausdorff limits of the sequence of incomplete Calabi-Yau metrics $(\mathcal{M}_{T},\omega_{T,CY})$ (scaled to fixed diameter) constructed in Theorem 6.3. As can be easily seen, the results are parallel to the statements in Theorem 1.1, and in Section 7 we shall not reproduce the arguments from here.

To begin with, we recall the notion of measured Gromov-Hausdorff convergence. We refer the readers to [CC97] for the general theory about this.

Definition 6.11 (Measured Gromov-Hausdorff convergence).

Let $(M_{j}^{m},g_{j},p_{j})$ be a sequence of Riemannian manifolds with $\Ric_{g_{j}}\geq-(m-1)$ such that

(6.136)

(M_{j}^{m},g_{j},p_{j})\xrightarrow{GH}(X_{\infty},d_{\infty},p_{\infty})

for some metric space $(X_{\infty},d_{\infty},p_{\infty})$ , then by passing to a subsequence, the renormalized measures

(6.137)

d\underline{\nu}_{j}\equiv\frac{\dvol_{g_{j}}}{\Vol_{g_{j}}(B_{1}(p_{j}))}

converge to a Radon measure $d\underline{\nu}_{\infty}$ on $X_{\infty}$ which is called the renormalized limit measure. The Gromov-Hausdorff convergence together with the convergence of the renormalzied measures is called the measured Gromov-Hausdorff convergence.

In the general context of collapsed sequences with Ricci curvature bounded from below, $d\underline{\nu}_{\infty}$ behaves quite differently from the Hausdorff measures on $X_{\infty}$ induced by the limiting metric $d_{\infty}$ . In our specific context, $d\underline{\nu}_{\infty}$ has an explicit form and it effectively reveals the geometric singularity information in the collapsing spaces.

Now return to our context. We are interesting in the measured Gromov-Hausdorff limits of $(\mathcal{M}_{T},\omega_{T},d\underline{\nu}_{T})$ , where $d\underline{\nu}_{T}=\frac{1}{n!}\omega_{T}^{n}$ is the volume measure of the metric $\omega_{T}$ . Using the error estimate in Proposition 4.23, the convergence is in fact dominated by large scale geometries of the neck metric $(\mathcal{M}_{T},\omega_{T})$ constructed in Section 4.1. So we shall only perform the calculation using the metrics $\omega_{T}$ , and the latter are fairly explicit by construction.

Gromov-Hausdorff limit:

By construction and direct calculation one sees that $g_{T}$ in large scale is approximated by the $1$ dimensional metric tensor $T^{\frac{(2-n)(n+1)}{n}}L_{T}(z)^{n-1}dz^{2}$ . In particular, the diameter is of order $T^{\frac{n+1}{n}}$ . This suggests rescaling the metric $g_{T}$ by $T^{-\frac{2(n+1)}{n}}$ in order to obtain bounded diameter. Indeed, upon the change of variable $z=T\cdot\xi$ , we see $T^{-\frac{2(n+1)}{n}}g_{T}$ converges to the one dimensional metric $(1+k_{\mp}\xi)^{n-1}d\xi^{2}$ , $\xi\in[-k_{-}^{-1},-k_{+}^{-1}]$ in the Gromov-Hausdorff sense.

The above limit can be transformed into the standard metric on the unit interval $(\mathbb{I},dv^{2})$ via a constant rescaling and the following coordinate change

(6.138)

\begin{cases}1+k_{+}(\frac{1}{k_{-}}-\frac{1}{k_{+}})v=(1+k_{+}\xi)^{\frac{n+1}{2}},\ \ \xi>0;\\ 1+k_{-}(\frac{1}{k_{-}}-\frac{1}{k_{+}})v=(1+k_{-}\xi)^{\frac{n+1}{2}},\ \ \xi<0.\end{cases}

Renormalized limit measure:

Again we first calculate by definition

(6.139)

\text{Vol}(\mathcal{M}_{a\leq z\leq b},\omega_{T})=\int_{a}^{b}dz\frac{T^{2-n}}{(n-1)!}\int_{D}\tilde{\omega}(z)^{n-1}.

Upon the change of variable $z=T\cdot\xi$ , this we get

(6.140)

\text{Vol}(\mathcal{M}_{a\leq\xi\leq b},\omega_{T})=T^{2}\int_{a}^{b}d\xi\int_{D}\frac{1}{(n-1)!}(1+k_{\pm}\xi)^{n-1}\omega_{D}^{n-1}.

So up to constant, the renormalized limit measure has density function given by $(1+\pm\xi)^{n-1}d\xi$ . Changing to the $v$ -variable this becomes (again up to constant multiplication)

(6.141)

d\underline{\nu}_{\infty}=\begin{cases}(\frac{v}{-k_{+}}+\frac{1}{k_{-}-k_{+}})^{\frac{n-1}{n+1}}dv,\ \ v\in[\frac{k_{+}}{k_{-}-k_{+}},0];\\ (\frac{v}{-k_{-}}+\frac{1}{k_{-}-k_{+}})^{\frac{n-1}{n+1}}dv,\ \ v\in[0,\frac{k_{-}}{k_{-}-k_{+}}].\end{cases}

Fibration structure:

There is an obvious fibration of $\mathcal{M}_{T}$ over $[T_{-},T_{+}]$ using the coordinate function $z$ . Composing with above coordinate changes, we obtain a fibration

(6.142)

\mathcal{F}_{T}:\mathcal{M}_{T}\rightarrow\mathbb{I};\bm{x}\mapsto v(\bm{x}).

It is clear that for any $v\neq 0$ , $\mathcal{F}_{T}^{-1}(v)$ is an $S^{1}$ bundle over $D$ , whose first Chern class is given by $c_{1}(L_{\pm})$ depending on the sign of $v$ , and $\mathcal{F}_{T}^{-1}(0)$ is an singular $S^{1}$ fibration over $D$ , with vanishing circles along $H$ .

Bubble classification:

From our analysis in Section 4.3, it is clear that suitable rescalings around the vanishing circles in $\mathcal{F}_{T}^{-1}(0)$ are given by the product space $\mathbb{C}_{TN}^{2}\times\mathbb{C}^{n-2}$ . Also suitable rescalings around the ends $z=T_{\pm}$ gives the incomplete Calabi model spaces.

We close this section by giving the following remarks regarding the regularity of the renormalized limit measure.

Remark 6.11.1.

It can be seen from the above formulae that the limiting density function $\mathscr{V}_{\infty}=\frac{d\underline{\nu}_{\infty}}{dv}$ is a Lipschitz function on $\mathbb{I}$ and it is smooth everywhere in the interior of $\mathbb{I}$ except at $v=0$ . On the other hand, the singular fiber of $\mathcal{F}$ precisely appears at $t=0$ . So in our context, the singularity of the renormalized limit measure $d\underline{\nu}_{\infty}$ effectively characterizes the singularity behavior of the collapsing geometry.

Remark 6.11.2.

By Cheeger-Colding (see [CC00], theorem 4.6), in the regular set $\mathcal{R}$ of a general Ricci-limit space, the density function $\mathscr{V}_{\infty}$ of the renormalized limit measure always exists and is Hölder continuous. Our example tells us that, in general, one cannot expect the regularity of $\mathscr{V}_{\infty}$ to be differentiable in $\mathcal{R}$ (even though $\mathcal{R}$ is a smooth Riemannian manifold). We thank Shouhei Honda for pointing this out.

Remark 6.11.3.

If we use rescale the metrics further around the point $v=0$ such that the sequence of spaces collapse to the complete real line $\mathbb{R}$ , then $d\underline{\nu}_{\infty}$ coincides with the standard Lebesgue measure. In particular, the singularity at $v=0$ disappears. This fact can be quickly seen by scaling-up the coordinates $v$ . This is compatible with the general theory of Ricci-limit spaces. That is, due to Cheeger-Colding, the renormalized limit measure always splits off the Lebesgue measure of $\mathbb{R}$ if the limit space isometrically splits off $\mathbb{R}$ (see proposition 1.35 in [CC97] for more details).

(6.8)		$\displaystyle\\|\sqrt{-1}\partial\bar{\partial}\phi\\|_{\mathfrak{S}_{1}}$	$\displaystyle\equiv\\|\sqrt{-1}\partial\bar{\partial}\phi\\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(X_{t})},$
(6.9)		$\displaystyle\\|f\\|_{\mathfrak{S}_{2}}$	$\displaystyle\equiv\\|\sqrt{-1}\partial\bar{\partial}\phi\\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(X_{t})}.$

(6.74)	$\displaystyle\\|\nabla v_{j}\\|_{C_{\delta,\nu+1,\mu}^{0}(\mathcal{M}_{j})}$	$\displaystyle=\\|\nabla u_{j}\\|_{C_{\delta,\nu+1,\mu}^{0}(\mathcal{M}_{j})}\leq 1,$
(6.75)	$\displaystyle\\|\nabla^{2}v_{j}\\|_{C_{\delta,\nu+2,\mu}^{0}(\mathcal{M}_{j})}$	$\displaystyle=\\|\nabla^{2}u_{j}\\|_{C_{\delta,\nu+2,\mu}^{0}(\mathcal{M}_{j})}\leq 1,$
(6.76)	$\displaystyle\\|\Delta v_{j}\\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(\mathcal{M}_{j})}$	$\displaystyle=\\|\Delta u_{j}\\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(\mathcal{M}_{j})}\to 0,$
(6.77)	$\displaystyle\\|v_{j}\\|_{C_{\delta,\nu,\mu}^{0}(\mathcal{M}_{j})}$	$\displaystyle\leq C_{0}.$

6. Perturbation to Calabi-Yau metrics on the neck [054R]

6. Perturbation to Calabi-Yau metrics on the neck

6.1. Framework of perturbation

Lemma 6.1 (Implicit function theorem).

Remark 6.1.1.

Lemma 6.2.

Proof.

Theorem 6.3 (Existence of S1S^{1}-invariant Calabi-Yau metrics).

Lemma 6.4 (Nonlinear error estimate).

Proof.

6.2. Some Liouville type theorems and removable singularity theorems

Lemma 6.5 (Removable singularity).

Proof.

Lemma 6.6 (Liouville theorem on ℝm+n\mathbb{R}^{m+n}).

Proof.

Lemma 6.7 (Liouville theorem on a cylinder).

6.3. Weighted analysis and existence of incomplete Calabi-Yau metrics

Proposition 6.8 (Uniform injectivity estimate on the neck).

Proposition 6.9 (Weighted Schauder estimate on the neck, the global version).

Proof.

Proposition 6.10 (Uniform injectivity estimate on the neck).

Proof.

Proof of Theorem 6.3.

6.4. Geometric singularity and normalized limit measure

Definition 6.11 (Measured Gromov-Hausdorff convergence).

Remark 6.11.1.

Remark 6.11.2.

Remark 6.11.3.

Theorem 6.3 (Existence of $S^{1}$ -invariant Calabi-Yau metrics).

Lemma 6.6 (Liouville theorem on $\mathbb{R}^{m+n}$ ).