Uniqueness of some Calabi-Yau metrics on $\mathbf{C}^n$

TL;DR

This paper proves the uniqueness, up to scaling and isometry, of certain Calabi-Yau metrics on complex Euclidean spaces with a specific tangent cone at infinity, and discusses potential generalizations.

Contribution

It establishes the uniqueness of specific Calabi-Yau metrics on ^n with a given tangent cone, extending understanding of their geometric structure.

Findings

Uniqueness of the Calabi-Yau metrics up to scaling and isometry.

Characterization of tangent cones at infinity for these metrics.

Discussion of potential generalizations to other manifolds.

Abstract

We consider the Calabi-Yau metrics on $\mathbf{C}^n$ constructed recently by Yang Li, Conlon-Rochon, and the author, that have tangent cone $\mathbf{C}\times A_1$ at infinity for the $(n-1)$-dimensional Stenzel cone $A_1$. We show that up to scaling and isometry this Calabi-Yau metric on $\mathbf{C}^n$ is unique. We also discuss possible generalizations to other manifolds and tangent cones.

Equations

$$({\omega }_0+\sqrt{-1}\partial \overline{\partial }\phi {)}^n={\omega }_0^n,$$

$$\displaystyle\omega=\sqrt{-1}\partial\overline{\partial}\Big{(}|z|^{2}+\gamma_{1}(R\rho^{-\alpha})r^{2}+\gamma_{2}(R\rho^{-\alpha})|z|^{2/d}\varphi(z^{-1/d}\cdot x)\Big{)}$$

$$\Omega =\frac{dz\wedge d{x}_2\wedge \dots \wedge d{x}_n}{{\partial }_{x_1}f}.$$

$$D{z}^{\prime }+D^{d}f(x^{\prime })=0,$$

$$D^{1-d}\tilde{z}+f(\tilde{x})=0,$$

$$\sqrt{-1}\partial \overline{\partial }(|\tilde{z}{|}^2+{\tilde{r}}^2),$$

$$z{\partial }_z=\tilde{z}{\partial }_{\tilde{z}},x_j{\partial }_{x_k}={\tilde{x}}_j{\partial }_{{\tilde{x}}_k}.$$

$$|z{\partial }_z{|}_{\omega },|x_j{\partial }_{x_k}{|}_{\omega }<CD,$$

$$\tilde{z}=K^{-1}(z-z_0),\tilde{x}=K^{-1}\cdot x,\tilde{r}=K^{-1}r.$$

$$K^{-d}(K\tilde{z}+z_0)+f(\tilde{x})=0,$$

$$\sqrt{-1}\partial \overline{\partial }(|\tilde{z}{|}^2+{\tilde{r}}^2),$$

$$z{\partial }_z=(K\tilde{z}+z_0)K^{-1}{\partial }_{\tilde{z}},x_j{\partial }_{x_k}={\tilde{x}}_j{\partial }_{{\tilde{x}}_k}.$$

$$|z{\partial }_z{|}_{K^{-2}\omega }<C{K}^{-1}D,|x_j{\partial }_{x_k}{|}_{K^{-2}\omega }<C.$$

$$\tilde{z}=z_0^{-1/d}(z-z_0),\tilde{x}=z_0^{-1/d}\cdot x,\tilde{r}=|z_0{|}^{-1/d}r.$$

$$z_0^{1/d-1}\tilde{z}+1+f(\tilde{x})=0,$$

$$1+f(\tilde{x})=0,$$

$$\displaystyle\sqrt{-1}\partial\overline{\partial}\Big{(}|\tilde{z}|^{2}+\varphi(\tilde{x})\Big{)}.$$

$$z{\partial }_z=z_0^{-1/d}(z_0^{1/d}\tilde{z}+z_0){\partial }_{\tilde{z}},x_j{\partial }_{x_k}={\tilde{x}}_j{\partial }_{{\tilde{x}}_k}.$$

$$|z{\partial }_z{|}_{\omega }<|z_0{|}^{1/d}C|z_0{|}^{-1/d}|z_0|<CD,|x_j{\partial }_{x_k}{|}_{\omega }<C|z_0{|}^{1/d}<C{D}^{1/d},$$

$$f=f_0+z{f}_1+\bar{z}{f}_{\bar{1}}+z^2{f}_2+{\bar{z}}^2{f}_{\bar{2}}+|z{|}^2{f}_{1\bar{1}},$$

$$\Delta f={\Delta }^{\prime }{f}_0+z{\Delta }^{\prime }{f}_1+\bar{z}{\Delta }^{\prime }{f}_{\bar{1}}+z^2{\Delta }^{\prime }{f}_2+{\bar{z}}^2{\Delta }^{\prime }{f}_{\bar{2}}+|z{|}^2{\Delta }^{\prime }{f}_{1\bar{1}}+f_{1\bar{1}},$$

$${\Delta }^{\prime }{f}_1={\Delta }^{\prime }{f}_{\bar{1}}={\Delta }^{\prime }{f}_2$$

$$f_{1\bar{1}}+{\Delta }^{\prime }{f}_0$$

$$c+{\Delta }^{\prime }{f}_0=0,$$

$$u=V(r^2)=V(|x{|}^{2\frac{n-2}{n-1}})=\frac{n-2}{n-1}|x{|}^{-\frac{2}{n-1}}{a}_{jk}{x}_j{\bar{x}}_k.$$

$$V_{\mathbf{a}}(|z{|}^2+|x{|}^{2\frac{n-2}{n-1}})=|x{|}^{-\frac{2}{n-1}}{a}_{jk}{x}_j{\bar{x}}_k.$$

$$W(|z{|}^2+|x{|}^{2\frac{n-2}{n-1}})=(n-1)|z{|}^2-|x{|}^{2\frac{n-2}{n-1}}.$$

$$V=\mathrm{Re}(z{\partial }_z+\frac{1}{2}{x}_i{\partial }_{x_i}),$$

$$L_V\Omega =\frac{n}{2}\Omega ,$$

$$V(|z{|}^2+|x{|}^{2\frac{n-2}{n-1}})-\frac{1}{2}(|z{|}^2+|x{|}^{2\frac{n-2}{n-1}})=\frac{1}{2}|z{|}^2-\frac{1}{2(n-1)}|x{|}^{2\frac{n-2}{n-1}},$$

$$V(|z{|}^2+|x{|}^{2\frac{n-2}{n-1}})-\beta (|z{|}^2+|x{|}^{2\frac{n-2}{n-1}})=h_{aut}.$$

Full text

1. Introduction

††The author is supported in part by NSF grants DMS-1350696 and DMS-1906216.

On a compact Kähler manifold with vanishing first Chern class, Yau’s solution of the Calabi conjecture [34] shows that any Kähler class admits a unique Calabi-Yau metric. In the non-compact setting there are many constructions of complete Calabi-Yau manifolds with different asymptotic behaviors by Cheng-Yau [6], Tian-Yau [32, 33] and others, and even fixing the Kähler class these metrics are typically not unique. To recover uniqueness, in general one needs to put conditions on the asymptotics of the metric. Our goal in this paper is to prove such a uniqueness result for certain Calabi-Yau metrics on $\mathbf{C}^{n}$.

The Taub-NUT metric on $\mathbf{C}^{2}$ is an example of a non-flat Kähler metric with the same volume form as the Euclidean metric (see LeBrun [18]). This metric does not have maximal volume growth and in fact the flat metric is the unique Ricci flat metric on $\mathbf{C}^{2}$ with maximal volume growth (see Tian [31]). It turns out that in higher dimensions this is no longer the case, and for $n\geq 3$, $\mathbf{C}^{n}$ admits a complete Calabi-Yau metric $\omega_{0}$ with tangent cone $\mathbf{C}\times A_{1}$ at infinity. Here $A_{1}$ is the $(n-1)$-dimensional $A_{1}$ singularity $x_{1}^{2}+\ldots+x_{n}^{2}=0$ equipped with the Stenzel cone metric (see Li [20], Conlon-Rochon [12] and the author’s work [30]). These metrics all have the same volume form as the Euclidean metric, and in fact there are infinitely many other metrics with the same volume form, and different tangent cones at infinity.

It is therefore natural to try to classify Calabi-Yau metrics with a prescribed tangent cone at infinity. Classification results have previously been obtained by Kronheimer [17] in the case of surfaces, and Conlon-Hein [10] in higher dimensions in the asymptotically conical setting, i.e. when the metric converges at a polynomial rate to a Ricci flat Kähler cone with smooth link. For instance in [10] the asymptotically conical Calabi-Yau manifolds with tangent cone $A_{1}$ are classified. Compared to these the main novelty in our work is that we are able to deal with tangent cones that do not have isolated singularities. Our main result is the following uniqueness statement for the metric $\omega_{0}$ on $\mathbf{C}^{n}$.

Theorem 1**.**

Suppose that $\omega$ is a complete Calabi-Yau metric on $\mathbf{C}^{n}$ with tangent cone $\mathbf{C}\times A_{1}$ at infinity. Then there is a biholomorphism $F:\mathbf{C}^{n}\to\mathbf{C}^{n}$ and a constant $a>0$ such that $\omega=aF^{*}\omega_{0}$.

We emphasize that for a Calabi-Yau manifold with maximal volume growth the tangent cone at infinity has a natural complex structure on the regular set (which extends in general to the singular set by the main results in [15, 24]). When we say that the tangent cone is $\mathbf{C}\times A_{1}$ we are requiring that the complex structures agree as well as the metric structures, since in principle there may be different complex structures on a given metric cone.

The proof of Theorem 1 can likely be extended to classify Calabi-Yau metrics on $\mathbf{C}^{n}$ with other tangent cones, as well as $\partial\bar{\partial}$-exact Calabi-Yau metrics on more general manifolds. We will discuss this in Section 5. The proof relies on two main ingredients. On the one hand, given a $\partial\bar{\partial}$-exact Calabi-Yau metric $(X,\omega)$ with tangent cone $C(Y)$, the work of Donaldson-Sun [15] gives an algebraic description of the ring of polynomial growth holomorphic functions on $(X,\omega)$ in terms of the coordinate ring of $C(Y)$. When $C(Y)=\mathbf{C}\times A_{1}$, and $X\cong\mathbf{C}^{n}$, then we can use this description to obtain an embedding $X\to\mathbf{C}^{n+1}$ as the hypersurface $z+x_{1}^{2}+\ldots+x_{n}^{2}=0$, such that the functions $z,x_{i}$ have degrees $1,\frac{n-1}{n-2}$ respectively. This is the basic input that allows us to compare the unknown metric $(X,\omega)$ with the reference metric $(\mathbf{C}^{n},\omega_{0})$, which is constructed by viewing $\mathbf{C}^{n}\subset\mathbf{C}^{n+1}$ as the same hypersurface. We discuss this in Section 3.

While we end up proceeding in a different way, heuristically the idea is that using such an embedding we can hope to find a biholomorphism $F:\mathbf{C}^{n}\to X$, such that $F^{*}\omega=\omega_{0}+\sqrt{-1}\partial\overline{\partial}\varphi$ satisfies the Monge-Ampère equation

[TABLE]

and in addition $\varphi$ has subquadratic growth in the sense that $r^{-2}\sup_{B(0,r)}|\varphi|\to 0$ as $r\to\infty$. In practice we are not able to do this, but if we could, we would then like to show that $F^{*}\omega=\omega_{0}$, in analogy with the uniqueness result of Conlon-Hein [11, Theorem 3.1] in the setting of asymptotically conical spaces. Instead we can only find a sequence of such biholomorphisms $F$ on larger and larger balls. The technical heart of the proof is Proposition 7, which roughly speaking says that if on some large $R$-ball we have a solution of (1.1) such that $R^{-2}\varphi$ is small, then on a smaller $\lambda R$-ball we can find an “equivalent” potential $\varphi^{\prime}$ such that $(\lambda R)^{-2}\varphi^{\prime}$ is even smaller. The proof of this result will take up most of Section 4. Iterating this, and letting $R\to\infty$, leads to Theorem 1.

Finally let us mention some related works for minimal hypersurfaces. Regarding the uniqueness of minimal hypersurfaces with prescribed tangent cone at infinity, Simon-Solomon [29] and Mazet [25] showed that minimal hypersurfaces in $\mathbf{C}^{n+1}$ that are asymptotic to certain Simons cones are essentially unique. At the same time, the works by Simon [28, 27], and more recently Colombo-Edelen-Spolaor [9], address the behavior of minimal submanifolds that are near to a cone with non-isolated singularities, which is also a key point in our case. While the details are very different, there are certainly similarities between our approach and theirs.

Acknowledgements

I would like to thank Nick Edelen and Gang Liu for insightful discussions, as well as Shih-Kai Chiu and Yang Li for helpful comments on an earlier draft of the paper.

2. The reference metric

In this section we give some preliminary results about the Calabi-Yau metrics on $\mathbf{C}^{n}$ constructed in [20, 12, 30]. We follow the approach from [30]. We suppose that $f(x_{1},\ldots,x_{n})$ is a polynomial such that $V_{0}=f^{-1}(0)\subset\mathbf{C}^{n}$ has an isolated normal singularity at the origin. We assume that $V_{0}$ admits a Calabi-Yau cone metric $\omega_{V_{0}}=\sqrt{-1}\partial\overline{\partial}r^{2}$, whose homothetic action is diagonal, with weights $(w_{1},\ldots,w_{n})$, and $f$ is homogeneous of degree $d>2$ under this action. The basic example we are concerned with is $f=x_{1}^{2}+\ldots+x_{n}^{2}$, in which case $w_{i}=\frac{n-1}{n-2}$, and we let $r^{2}=|x|^{2\frac{n-2}{n-1}}$. In general it follows from Conlon-Hein [11] (see also [30, Section 2]) that the smoothing $V\subset\mathbf{C}^{n}$ given by the equation $1+f(x)=0$ admits a Calabi-Yau metric $\omega_{V_{1}}=\sqrt{-1}\partial\overline{\partial}\varphi(x)$, with tangent cone $V_{0}$ at infinity.

We then consider the hypersurface $X\subset\mathbf{C}\times\mathbf{C}^{n}$ given by $z+f(x)=0$, which is biholomorphic to $\mathbf{C}^{n}$. The main result of [30] is that there exists a Calabi-Yau metric $\omega_{0}$ on $\mathbf{C}^{n}$ with tangent cone $X_{0}=\mathbf{C}\times V_{0}$ at infinity, which is uniformly equivalent to the metric

[TABLE]

outside a compact set. Here $\gamma_{i}(s)$ are suitable cutoff functions such that $\gamma_{1}+\gamma_{2}=1$, $\gamma_{1}$ is supported where $s>0$ while $\gamma_{2}$ is supported where $s<2$; the function $R$ is such that $\sqrt{-1}\partial\overline{\partial}R^{2}$ defines a cone metric on $\mathbf{C}^{n}$ with the same homothetic action as $V_{0}$; the function $\rho^{2}=|z|^{2}+R^{2}$; $\alpha\in(1/d,1)$ and $z^{-1/d}\cdot x$ is defined using the homothetic action, choosing a branch of $\log$. The form $\omega$ defines a metric when restricted to $X$ outside of a compact set, and the Calabi-Yau metric $\omega_{0}$ that is constructed is asymptotic to $\omega$ at infinity, in the sense that $|\omega_{0}-\omega|_{\omega}\to 0$. The volume form of $\omega_{0}$ is $\sqrt{-1}^{n^{2}}\Omega\wedge\bar{\Omega}$ for

[TABLE]

For more details see [30].

From [30, Proposition 9] we have the following. For large $D$, we can consider a new embedding $X\to\mathbf{C}^{n+1}$ by the functions $z^{\prime}=D^{-1}z,x_{i}^{\prime}=D^{-w_{i}}x_{i}$. The image has equation

[TABLE]

i.e. $D^{1-d}z^{\prime}+f(x^{\prime})=0$, recalling that $f$ has degree $d$ under the homothetic action. We equip this hypersurface $X^{\prime}$ with the scaled down metric $D^{-2}\omega_{0}$. Here, and below, let us denote by $\Psi(\epsilon)$ a function converging to zero as $\epsilon\to 0$. This function may change from line to line. From [30, Proposition 9] we see that there is a constant $\theta<\Psi(D^{-1})$ satisfying the following. We define the map $G:B_{X^{\prime}}(0,1)\to X_{0}$ using the nearest point projection on the set where $|x^{\prime}|>\theta$, and projection onto the $z$-axis where $|x^{\prime}|\leq\theta$. Then $G$ is a $\Psi(D^{-1})$-Gromov-Hausdorff approximation to $B_{X_{0}}(0,1)$. One useful consequence of this is that the distance from the origin in $(X,\omega_{0})$ is uniformly equivalent to the function $\rho$. We will need the following.

Proposition 2**.**

• (a)

The holomorphic functions $z,x_{i}$ on $(X,\omega_{0})$ have polynomial growth with degrees $d(z)=1,d(x_{i})=w_{i}$.

• (b)

Consider the special case $f=x_{1}^{2}+\ldots+x_{n}^{2}$. Then the vector fields $2z\partial_{z}+x_{i}\partial_{x_{i}}$ and $a_{jk}x_{j}\partial_{x_{k}}$ for skew-symmetric $(a_{jk})$ all have at most linear growth.

Proof.

The statement in (a) is immediate from the fact that the distance function is uniformly equivalent to $\rho$.

For part (b), we can work using the description of the metric $\omega$ in regions I–V in the proof of [30, Proposition 5] (note that $\omega$ is uniformly equivalent to $\omega_{0}$). In each region we choose new coordinates in which we have a good model for the form $\omega$ and so we can bound our vector fields. Let us consider regions I, III and V, the others being very similar.

Region I: Here $R>\kappa\rho$ for some fixed small $\kappa>0$, and we assume $\rho\in(D/2,2D)$ for $D$, which will then be uniformly equivalent to the distance from the origin. We change coordinates to $\tilde{z}=D^{-1}z$ and $\tilde{x}=D^{-1}\cdot x$, and we let $\tilde{r}=D^{-1}r$. In these coordinates $X$ has equation

[TABLE]

and in the proof of [30, Proposition 5] the scaled down metric $D^{-2}\omega$ on this hypersurface is compared to the product metric

[TABLE]

on the hypersurface with equation $f(\tilde{x})=0$ (i.e. the product $X_{0}$). Since $|\tilde{z}|<2$ and $\tilde{r}\in(\kappa/2,4\kappa)$, as well as $d>1$, as $D\to\infty$ then in these coordinates the hypersurface (2.4) converges smoothly to $X_{0}$. Because of this we can compute the norms of our vector fields with respect to the metric (2.5). For this we have

[TABLE]

The norms of these vector fields are uniformly bounded for the metric in (2.5), which is uniformly equivalent to $D^{-2}\omega$ (under identifying the two hypersurfaces), and so

[TABLE]

for a constant $C$.

Region III: Here $R\in(K/2,2K)$, and $K\in(\rho^{\alpha},2\rho^{\alpha})$. We suppose $\rho\in(D/2,2D)$, so $|z|$ is comparable to $D$. We choose a fixed $z_{0}$ such that $|z-z_{0}|<K$, and we change variables as follows:

[TABLE]

In these coordinates $X$ is given by the equation

[TABLE]

and we compare again to the product metric

[TABLE]

on the hypersurface $f(\tilde{x})=0$. We have $|\tilde{z}|<1$, $\tilde{R}\in(1/2,2)$, and $K^{-d}z_{0}\to 0$ as $K\to\infty$, since $K^{d}\gg D$. This means that as $K,D\to\infty$, we can measure the norms of our vector fields on $X_{0}$ with the product metric. We have

[TABLE]

It follows that

[TABLE]

Since $D$ is comparable to the distance from the origin, and $K\ll D$, this implies the estimate we want.

Region V: Here $R<2\kappa^{-1}\rho^{1/d}$, $\rho\in(D/2,2D)$, so $|z|$ is comparable to $D$. We choose a fixed point $z_{0}$ with $|z-z_{0}|<D^{1/d}$, so we also have $|z_{0}|\sim D$. We scale our metric down by a factor of $|z_{0}|^{1/d}$, and change coordinates by

[TABLE]

We have $|\tilde{z}|,\tilde{r}<C$ for a uniform $C$. The equation of $X$ is

[TABLE]

and as $D\to\infty$, this converges to hypersurface with equation

[TABLE]

and the metric $|z_{0}|^{-2/d}\omega$ converges to

[TABLE]

We have

[TABLE]

The norms of these vector fields are uniformly bounded with respect to (2.16), and so scaling back up, we have

[TABLE]

which gives the required bound. ∎

2.1. Subquadratic harmonic functions on $\mathbf{C}\times A_{1}$

Let us consider the tangent cone $C(Y)=\mathbf{C}\times A_{1}$ embedded in $\mathbf{C}\times\mathbf{C}^{n}$ as the hypersurface $x_{1}^{2}+\ldots+x_{n}^{2}=0$, and equipped with the Stenzel cone metric $\sqrt{-1}\partial\overline{\partial}(|z|^{2}+|x|^{2\frac{n-2}{n-1}})$. We need to understand the harmonic functions on $C(Y)$ with at most quadratic growth. In Hein-Sun [16] a general result is given on Calabi-Yau cones with isolated singularities, saying that the strictly subquadratic harmonic functions are all pluriharmonic (this was first used crucially in Conlon-Hein [11]), while the space of exactly quadratic growth harmonic functions decomposes as the sum of pluriharmonic functions and harmonic functions that arise from isometries of the link. See also Chiu [7] for results in the case of more singular cones. We have the following.

Lemma 3**.**

The space $\mathcal{H}_{\leq 2}$ of real harmonic functions on $C(Y)$ with at most quadratic growth are given by linear combinations of the following:

1. (1)

the real and imaginary parts of $1,z,z^{2},x_{i}$, 2. (2)

the function $(n-1)|z|^{2}-|x|^{2\frac{n-2}{n-1}}$, 3. (3)

the functions $|x|^{-\frac{2}{n-1}}a_{jk}x_{j}\bar{x}_{k}$, where $(a_{jk})\in\sqrt{-1}\mathfrak{o}(n,\mathbf{R})$ is a purely imaginary complex orthogonal matrix.

Proof.

A general approach to this result is to extend Hein-Sun [16, Theorem 2.14] to singular tangent cones. This can be done along the lines of the work in Chiu [7], using cutoff functions to justify the required integration by parts near the singular set.

Alternatively we can follow the approach from [30, Corollary 12] using the Fourier transform in the $\mathbf{C}$-direction to analyze harmonic functions on the product $\mathbf{C}\times A_{1}$. The conclusion from this approach is that any harmonic function $f$ of at most quadratic growth can be written as

[TABLE]

for functions $f_{0},f_{1},f_{\bar{1}},f_{2},f_{\bar{2}},f_{1\bar{1}}$ on the cone $A_{1}$. We have

[TABLE]

where $\Delta^{\prime}$ is the Laplacian on $A_{1}$. It follows from $\Delta f=0$ that

[TABLE]

In addition since $f$ has at most quadratic growth, $f_{1},f_{\bar{1}},f_{2},f_{\bar{2}}$ are all subquadratic harmonic functions, so by [16, Theorem 2.14] they are pluriharmonic. Since the non-constant holomorphic functions on $A_{1}$ have faster than linear growth, these functions must all be constant. The function $f_{1\bar{1}}$ is harmonic, and $|z|^{2}f_{1\bar{1}}$ has at most quadratic growth. It follows that $f_{1\bar{1}}=c$ is constant. Then

[TABLE]

so $f_{0}^{\prime}=(n-1)f_{0}-c|x|^{2\frac{n-2}{n-1}}$ is harmonic, and has at most quadratic growth. Using [16, Theorem 2.14] again, we have that $f_{0}^{\prime}$ is a linear combination of real and imaginary parts of $1,x_{i}$, and functions $u$ such that $V=\nabla u$ is a real holomorphic vector field on $A_{1}$ commuting with $r\partial_{r}$ such that $JV(r)=0$. We then have $V=\mathrm{Re}(a_{jk}x_{j}\partial_{x_{k}})$ for $a_{jk}$ a purely imaginary skew symmetric matrix. Using the identity $\sqrt{-1}\partial\overline{\partial}u=L_{\nabla u}\sqrt{-1}\partial\overline{\partial}r^{2}=\sqrt{-1}\partial\overline{\partial}V(r^{2})$, up to adding a pluriharmonic function to $u$, we have

[TABLE]

The result follows from this. ∎

The functions in (1) are all the pluriharmonic functions of at most quadratic growth, while (2) and (3) correspond to automorphisms of $C(Y)$ commuting with the homothetic scaling, which has weights $(1,\frac{n-1}{n-2},\ldots,\frac{n-1}{n-2})$ on $(z,x_{1},\ldots,x_{n})$. As in the proof, the functions $|x|^{-\frac{2}{n-1}}a_{jk}x_{j}\bar{x}_{k}$ correspond to the vector fields $V_{\mathbf{a}}=\frac{n-1}{n-2}\mathrm{Re}(a_{jk}x_{j}\partial_{x_{k}})$, in the sense that

[TABLE]

These vector fields preserve the hypersurfaces $cz+x_{1}^{2}+\ldots+x_{n}^{2}=0$ for all $c$, and the volume form $\Omega$.

Similarly, the function $(n-1)|z|^{2}-|x|^{2\frac{n-2}{n-1}}$ corresponds to the real holomorphic vector field $W=\mathrm{Re}((n-1)z\partial_{z}-\frac{n-1}{n-2}x_{i}\partial_{x_{i}})$, i.e.

[TABLE]

This vector field $W$ preserves $C(Y)\subset\mathbf{C}^{n+1}$, however it does not preserve the hypersurfaces $cz+x_{1}^{2}+\ldots+x_{n}^{2}=0$. Instead we let

[TABLE]

which does preserve all of these hypersurfaces. The vector field $V$ satisfies

[TABLE]

and

[TABLE]

which is a scalar multiple of the function in (2). We conclude the following.

Lemma 4**.**

Suppose that $h$ is a harmonic function on $C(Y)$ with at most quadratic growth, and write $h=h_{ph}+h_{aut}$, where $h_{ph}$ is in the span of the type (1) functions in Lemma 3, and is pluriharmonic, while $h_{aut}$ is in the span of the type (2) and (3) functions.

We can find a real holomorphic vector field $V$ preserving the hypersurfaces $cz+x_{1}^{2}+\ldots+x_{n}^{2}=0$, and a constant $\beta$ such that $L_{V}\Omega=n\beta\Omega$, and

[TABLE]

In addition we have $|\beta|\leq C\|h\|$ and $V=a_{0}z\partial_{z}+a_{jk}x_{j}\partial_{x_{k}}$ with $|a_{0}|,|a_{jk}|\leq C\|h\|$, for a constant $C$, where $\|h\|$ denotes the $L^{2}$ norm on $B(0,1)\subset C(Y)$.

3. Special embeddings

In this section $(X,\eta)$ is a complete Calabi-Yau manifold such that $X$ is biholomorphic to $\mathbf{C}^{n}$, and $X$ has tangent cone $C(Y)=\mathbf{C}\times A_{1}$ at infinity. Let us fix a basepoint $p\in X$, and denote by $B_{i}$ the ball $B(p,2^{i})$, with the metric $2^{-2i}\eta$, so that $B_{i}$ is a unit ball in the scaled down metric. By assumption, the sequence $B_{i}$ converges to the unit ball $B(0,1)\subset C(Y)$ in the Gromov-Hausdorff sense. We will view $C(Y)\subset\mathbf{C}^{n+1}$ as defined by the equation $x_{1}^{2}+\ldots+x_{n}^{2}=0$, in terms of the coordinates $z,x_{i}$ on $\mathbf{C}^{n+1}$ . The cone $C(Y)$ is equipped with the Ricci flat Stenzel metric given by

[TABLE]

which has volume form $\sqrt{-1}^{n^{2}}\Omega\wedge\bar{\Omega}$, in terms of the $\Omega$ from (2.2). The main result of this section is the following.

Proposition 5**.**

There is a sequence of holomorphic embeddings

[TABLE]

with the following properties:

1. (1)

the image $F_{i}(X)$ is given by the equation

[TABLE]

for some $a_{i}>0$, 2. (2)

the volume form $\eta^{n}$ satisfies

[TABLE] 3. (3)

$a_{i}/a_{i+1}\to 2^{n/(n-2)}$* as $i\to\infty$,* 4. (4)

on the ball $B_{i}$ the map $F_{i}$ gives a $\Psi(i^{-1})$-Gromov-Hausdorff approximation to the embedding $B(0,1)\to\mathbf{C}^{n+1}$. More precisely, we have a $\Psi(i^{-1})$-Gromov-Hausdorff approximation $g:B_{i}\to B(0,1)$, such that $|F_{i}-g|<\Psi(i^{-1})$ on $B_{i}$. Recall that here $B(0,1)\subset\mathbf{C}^{n+1}$ is the unit ball of $C(Y)$ under our embedding, and $\Psi(i^{-1})$ denotes a function converging to zero as $i\to\infty$.

The main input for this result is the work of Donaldson-Sun [15] on the algebro-geometric study of tangent cones, and we first review the results that we use.

3.1. Donaldson-Sun theory

In [14, 15], Donaldson-Sun consider non-collapsed Gromov-Hausdorff limits of compact polarized Kähler manifolds with bounded Ricci curvature. We observe that for many of the arguments compactness is not required (see also Liu [21, 22] for related work in the non-compact setting). More precisely, suppose that $(M_{i},L_{i},\omega_{i},p_{i})$ is a sequence of complete pointed $n$-dimensional Kähler manifolds with line bundles $L_{i}\to M_{i}$ equipped with Hermitian metrics with curvature $-\sqrt{-1}\omega_{i}$. In addition suppose that we have the Einstein condition $\mathrm{Ric}(\omega_{i})=\lambda_{i}\omega_{i}$ with $|\lambda_{i}|\leq 1$, and the non-collapsing condition $\mathrm{Vol}(B(p_{i},1))>\kappa>0$ for all $i$, for a fixed $\kappa>0$. If in addition we were to assume that the manifolds were compact, then the sequence would be in the class $\mathcal{K}(n,\kappa)$ considered in [15].

Let us suppose that $(Z,p)$ is the pointed Gromov-Hausdorff limit of the sequence $(M_{i},\omega_{i},p_{i})$. Then [15, Theorems 1.1, 1.2, 1.3] hold, i.e. $Z$ has the structure of a normal complex analytic space, and tangent cones to $Z$ have the structure of affine varieties and are unique. To see this note that the basic construction in [14] that is used in the arguments is to “graft” a holomorphic function from a tangent cone to $Z$, using cutoff functions, onto $M_{i}$ for sufficiently large $i$, and then use the Hörmander $L^{2}$-estimate the perturb the resulting approximately holomorphic section of (a power of) $L_{i}$ to a holomorphic section $s$. The grafting is a local construction, and the Hörmander estimate holds on complete Kähler manifolds (see e.g. [13, Theorem 4.5]). Finally one uses Moser iteration and Bochner-Weitzenbock type formulas to bound the $L^{\infty}$ norms of $s$ and $\nabla s$ in terms of the $L^{2}$-norm of $s$ (see [14, Proposition 2.1]). Since under the non-collapsing condition and Ricci curvature lower bound we can control the Sobolev constant on geodesic balls (see e.g. Anderson [2, Theorem 4.1]), the same estimates hold in our setting.

We need to use the results in [15, Section 3.4], however the assumptions there are that the limit space $(Z,p)$ is a scaled limit of a sequence in $\mathcal{K}(n,\kappa)$, with scaling factors tending to infinity. We claim, however, that the same results hold for a complete Calabi-Yau manifold $(M,\omega)$, where $\omega=\sqrt{-1}\partial\overline{\partial}\psi$ for a global Kähler potential $\psi$, which has maximal volume growth: $\mathrm{Vol}(B(p,r))>\kappa r^{2n}$ for all $r>0$, for a basepoint $p\in M$. The basic reason is that in this case the tangent cone at infinity is still the Gromov-Hausdorff limit of a sequence of polarized Kähler manifolds as above: for any sequence $\lambda_{i}\to 0$, we can consider the sequence $(M_{i},\omega_{i},L_{i},p_{i})$, where $M_{i}=M,\omega_{i}=\lambda_{i}^{2}\omega,p_{i}=p$, and $L_{i}$ is the trivial bundle equipped with the metric $e^{-\lambda_{i}^{2}\psi}$. Up to choosing a subsequence, this sequence converges in the Gromov-Hausdorff sense to a tangent cone at infinity $C(Y)$ of $(M,\omega)$. Using this, the arguments in [15, Section 2.2] can be applied to the limit space $C(Y)$ (instead of $(Z,p)$ in the statements of the propositions there) without any changes. In particular [15, Proposition 2.9] holds, showing that holomorphic functions on a ball in $C(Y)$ can be approximated by holomorphic functions on suitable balls in $M_{i}$. This is a crucial ingredient in [15, Proposition 3.26], which leads to the algebro-geometric description of the tangent cone $C(Y)$ in terms of the ring of polynomial growth holomorphic functions on $(M,\omega)$.

Let us briefly recall the results that we need from [15, Section 3.4], where for us $M$ plays the role of $Z$ there. If $R(M)$ denotes the ring of polynomial growth holomorphic functions on $M$, then $R(M)$ has a filtration

[TABLE]

Here $I_{k}$ is the space of polynomial growth holomorphic functions on $M$ with degree at most $d_{k}$, where $0=d_{0}<d_{1}<\ldots$ are the possible growth rates. For a holomorphic function $f$ on $X$, the growth rate $d(f)$ is defined by

[TABLE]

and $f$ has polynomial growth if $d(f)<\infty$. These growth rates are the same as the possible growth rates on the tangent cone $C(Y)$, and the dimensions $\dim I_{k}$ are equal to the corresponding dimensions on $C(Y)$. Let us write $R_{d_{k}}$ for the functions of degree $d_{k}$ on $C(Y)$, and $\mu_{k}=\dim R_{d_{k}}$.

By [15, Proposition 3.26] we can find decompositions $I_{k}=I_{k-1}\oplus J_{k}$, where $\dim J_{k}=\mu_{k}$, and $J_{k}$ admits an adapted sequence of bases. This means that, for fixed $k$, we have a sequence of bases $\{G^{i}_{1},\ldots,G^{i}_{\mu_{k}}\}$ for $J_{k}$, satisfying

1. (1)

$\|G^{i}_{a}\|_{B_{i}}=1$ for all $a$, and for $a\neq b$ we have $\lim_{i\to\infty}\int_{B_{i}}G_{a}^{i}\overline{G_{b}^{i}}=0$. Here $\|\cdot\|_{B_{i}}$ denotes the $L^{2}$-norm on $B_{i}$, and as above, $B_{i}$ is the ball $B(p,2^{i})$ scaled down to unit size. 2. (2)

$G_{a}^{i+1}=\mu_{ia}G_{a}^{i}+p^{i}_{a}$ for scalars $\mu_{ia}$, and $p^{i}_{a}\in\mathbf{C}\langle G^{i}_{1},\ldots G^{i}_{a-1}\rangle$, with $\|p^{i}_{a}\|_{B_{i}}\to 0$ as $i\to\infty$. 3. (3)

$\mu_{ia}\to 2^{d_{k}}$ as $i\to\infty$.

Suppose now that the coordinate ring $R(C(Y))$ is generated by $\bigoplus_{k\leq k_{0}}R_{d_{k}}$. It follows then that $R(M)$ is generated by $\bigoplus_{k\leq k_{0}}J_{k}$, and the adapted bases of $J_{k}$ for $k\leq k_{0}$ define embeddings $F_{i}:M\to\mathbf{C}^{N}$. Furthermore, under the Gromov-Hausdorff convergence $B_{i}\to B(0,1)\subset C(Y)$, the maps $F_{i}$ converge to an embedding $B(0,1)\to\mathbf{C}^{N}$ by an $L^{2}$-orthonormal basis of $R_{d_{0}}\oplus\ldots\oplus R_{d_{k_{0}}}$.

3.2. Proof of Proposition 5

We now specialize to the setting of Proposition 5. It will be helpful to write down the homogeneous holomorphic functions of low degree on $C(Y)=\mathbf{C}\times A_{1}$. Note that they are all spanned by polynomials in $z,x_{i}$, and $d(z)=1,d(x_{i})=\frac{n-1}{n-2}$. We treat three cases separately:

• •

$n=3$. In this case we have

[TABLE]

where in $R_{4}$ one term is redundant because of the equation $x_{1}^{2}+\ldots+x_{n}^{2}=0$.

• •

$n=4$. Here we have

[TABLE]

where again one term in $R_{3}$ is redundant.

• •

$n>4$.

[TABLE]

where again one term in $R_{\frac{2n-2}{n-2}}$ is redundant.

For simpicity we focus on the case $n=3$. The discussion in the other cases is completely analogous. The ring $R(C(Y))$ is generated by $R_{1}\oplus R_{2}$, and so by the results of Donaldson-Sun [15] discussed above, $R(X)$ is generated by $I_{2}=J_{0}\oplus J_{1}\oplus J_{2}$. This space of holomorphic functions on $X$ with at most quadratic growth has $\dim I_{2}=6$, and admits a sequence of adapted bases $\{G^{i}_{1},\ldots,G^{i}_{6}\}$. The growth rates of the functions are $d(G^{i}_{1})=0,d(G^{i}_{2})=1$ and $d(G^{i}_{j})=2$ for $j=3,4,5,6$, and for each $i$ we obtain an embedding

[TABLE]

with components $G^{i}_{j}$. On the balls $B_{i}$ these maps converge in the Gromov-Hausdorff sense to an embedding $F_{\infty}:B(0,1)\to\mathbf{C}^{6}$. The map $F_{\infty}$ is given by functions on $C(Y)$ with the degrees specified above, that are orthonormal on $B(0,1)$. Up to a unitary transformation commuting with the homothetic scaling (which has degrees $(0,1,2,2,2,2)$) we can assume that $F_{\infty}=(1,z,z^{2},x_{1},x_{2},x_{3})$. We can modify our sequence of adapted bases by the same unitary transformation, so that we still have $F_{i}\to F_{\infty}$ as $i\to\infty$.

Since $I_{0}$ consists of just the constants, the first component of $F_{i}$ is constant. In addition, we have $\|(G^{i}_{2})^{2}-G^{i}_{3}\|_{B_{i}}\to 0$ by the convergence of $F_{i}$ to $F_{\infty}$, while also $(G^{i}_{2})^{2}-G^{i}_{3}\in I_{2}$. It follows that $G^{i}_{3}=(G^{i}_{2})^{2}+\sum_{a}q_{ia}G^{i}_{a}$, where $|q_{iq}|<\Psi(i^{-1})$. Therefore dropping the first and third components of $F_{i}$, we still obtain embeddings $F_{i}^{\prime}:X\to\mathbf{C}^{4}$. Let us now fix $i$, and abusing notation, let us denote by $z,x_{1},x_{2},x_{3}$ the pullbacks under $F_{i}^{\prime}$ to $X$ of the coordinate functions on $\mathbf{C}^{4}$. By construction we have $d(z)=1,d(x_{i})=2$. The 20 functions

[TABLE]

are all in the space $I_{4}$ of at most quartic growth functions on $X$, but by the earlier discussion $\dim I_{4}=19$. Therefore we have one linear dependency between them, which determines the equation $f_{i}$ in $\mathbf{C}^{4}$ defining $F_{i}^{\prime}(X)$. Because of the Gromov-Hausdorff convergence of the $F_{i}^{\prime}$ to the embedding of $B(0,1)$ satisfying $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=0$, the equation $f_{i}(z,x_{1},x_{2},x_{3})=0$ has to be a perturbation of this equation. We can apply a linear transformation in $x_{1},x_{2},x_{3}$ that is $\Psi(i^{-1})$-close to the identity, to transform the quadratic expression in the $x_{j}$ that appears in $f_{i}$ to the quadric $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}$. Next we can complete the square in the $x_{j}$, applying changes of coordinates of the form $x_{j}\mapsto x_{j}+a_{j}z^{2}+b_{j}z+c_{j}$ with small $a_{j},b_{j},c_{j}$, to eliminate the terms of the form $x_{j},zx_{j},z^{2}x_{j}$ in $f_{i}$. We have now reduced our equation to one of the form

[TABLE]

where $\tilde{f}_{i}(z)$ is a quartic polynomial in $z$ with coefficients of order $\Psi(i^{-1})$. Since $X$ is biholomorphic to $\mathbf{C}^{3}$, $\tilde{f}_{i}$ must actually be linear, so $f_{i}(z)=d_{i}z+e_{i}$ for $|d_{i}|,|e_{i}|<\Psi(i^{-1})$ with $d_{i}\neq 0$. We can assume that $d_{i}>0$ by multiplying $z$ by a unit complex number.

So far we have only performed coordinate changes $\Psi(i^{-1})$-close to the identity, and so the new maps $F_{i}^{\prime\prime}:X\to\mathbf{C}^{4}$ that we obtain still converge on $B_{i}$, in the Gromov-Hausdorff sense, to our standard embedding $B(0,1)\to\mathbf{C}^{4}$. The image $F_{i}^{\prime\prime}(X)$ satisfies the equation

[TABLE]

We want to perform a further coordinate change $z\mapsto z+d_{i}^{-1}e_{i}$, so that the equation reduces to $d_{i}z+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=0$, however this may not be a small coordinate change, since we do not yet have information about the relative sizes of $d_{i},e_{i}$. For this we need the following.

Lemma 6**.**

Suppose that we have polynomial growth holomorphic functions $w,y_{1},y_{2},y_{3}$ on $X$ such that $d(w)=1$ and $d(y_{j})=2$, and

[TABLE]

for some $c\neq 0$. Then there exists a point $o\in X$ such that $w(o)=y_{j}(o)=0$, and moreover this point is independent of the choice of functions $w,y_{j}$ as above.

Proof.

In the above discussion we have already constructed functions $z,x_{j}$ satisfying

[TABLE]

Let us denote by $o\in X$ the point where $z(o)=x_{i}(o)=0$. In terms of these functions, because of the degree restrictions, we must have

[TABLE]

By scaling we can assume that $c=c^{\prime}=1$ to simplify notation. We have

[TABLE]

By our dimension counts earlier, up to a scalar multiple, there is only one linear equation satisfied by the functions

[TABLE]

namely $z+\mathbf{x}^{T}\mathbf{x}=0$. Therefore if we also have $w+\mathbf{y}^{T}\mathbf{y}=0$, then from (3.17) we first find that $A^{T}A$ is a multiple of the identity, and in particular $A$ is invertible. It then follows that we have $\mathbf{c}=\mathbf{d}=\mathbf{e}=0$, and in turn $b=0$. Therefore $w(o)=y_{j}(o)=0$, and conversely $o$ is the unique point where $w,y_{j}$ all vanish. ∎

Let us return to the proof of Proposition 5, and the map $F_{i}^{\prime\prime}:X\to\mathbf{C}^{4}$ whose image satisfies the equation

[TABLE]

By the Lemma above we have $z(o)=-d_{i}^{-1}e_{i}$, however since $F_{i}$ converges to the embedding $B(0,1)\to\mathbf{C}^{4}$, we also have $F_{i}(o)\to 0$. For this note that under the rescaled metric in the ball $B_{i}$ (centered at $p$) the point $o$ is contained in a ball of radius $C2^{-i}$ around $p$ where $C=\mathrm{dist}_{\eta}(p,o)$. In particular we have $d_{i}^{-1}e_{i}\to 0$, and so we can perform a final small coordinate change to reduce to the case $e_{i}=0$.

Next we consider the volume forms. The pullback $(F_{i}^{\prime\prime})^{*}(\Omega)$ defines a polynomial growth, nowhere vanishing holomorphic volume form on $X$. By the Calabi-Yau condition on $\eta$

[TABLE]

where $g_{i}$ is a nowhere vanishing polynomial growth holomorphic function on $X$. This means $g_{i}$ can be written as a polynomial in $x_{1},x_{2},x_{3}$, and therefore it is constant since it has no zeros. By the volume convergence under Gromov-Hausdorff convergence [8], remembering that we are using the scaled down metric $2^{-2i}\eta$ in the convergence, we find that $|g_{i}|^{2}2^{6i}\to 1$ as $i\to\infty$. Scaling $z$ by a factor $\Psi(i^{-1})$-close to 1, we can arrange that $|g_{i}|^{2}=2^{-6i}$.

Finally we address the claim that $a_{i}/a_{i+1}\to 8$. Let us denote the components of our maps $F_{i}^{\prime\prime}$ by $z_{i},x_{i,1},x_{i,2},x_{i,3}$, so that

[TABLE]

with $a_{i}>0$. From the proof of Lemma 6 we know that

[TABLE]

for a constant $c_{i}$ and a complex orthogonal matrix $A_{i}$. Using that $d(z_{i})=1,d(x_{i,j})=2$, and that the $z_{i},x_{i,j}$ have normalized $L^{2}$-norm approximately equal to 1 on $B_{i}$, we have $c_{i}\to 2^{-1}$ and $\sqrt{a_{i}^{-1}a_{i+1}c_{i}}\to 2^{-2}$ as $i\to\infty$. It follows that $a_{i}^{-1}a_{i+1}\to 2^{-3}$.

The cases $n=4$ and $n>4$ can be treated in an almost identical way, except in both cases we have $\dim I_{2}=5$, so we do not have to worry about the function $z^{2}$. In addition in the equation analogous to (3.12), the degree of $\tilde{f}_{i}$ is at most cubic when $n=4$, and at most quadratic when $n>4$. We leave the details to the reader.

4. Decay of the Kähler potential

In this section, we let $\omega=c^{2}\omega_{0}$ be a scaled down copy of the reference Calabi-Yau metric on $\mathbf{C}^{n}$, and we denote by $o\in(\mathbf{C}^{n},\omega)$ the origin. For a given $\epsilon>0$ we will assume that $c$ is sufficiently small so that $d_{GH}(B(o,\epsilon^{-1}),B(0,\epsilon^{-1}))<\epsilon$, where $0\in C(Y)$ is the origin in the cone $C(Y)=\mathbf{C}\times A_{1}$. Since $C(Y)$ is the tangent cone at infinity, this condition is equivalent to the Gromov-Hausdorff distance $d_{GH}(B(o,1),B(0,1))$ being sufficiently small. As in Section 2 we have an embedding $F_{c}:\mathbf{C}^{n}\to\mathbf{C}^{n+1}$ using the functions $cz,c^{\frac{n-1}{n-2}}x_{i}$, with image given by the equation

[TABLE]

In addition the maps $F_{c}$ on $B(o,1)$ converge in the Gromov-Hausdorff sense to the standard embedding $B(0,1)\to\mathbf{C}^{n+1}$ of the unit ball in $C(Y)$ as $c\to 0$.

The main technical result of the paper is the following.

Proposition 7**.**

There are $\lambda_{0},\alpha$ such that if $\lambda<\lambda_{0}$, and $\epsilon$ is sufficiently small (depending on $\lambda$), then we have the following. Suppose that $c$ above is small enough so that $d_{GH}(B(o,\epsilon^{-1}),B(0,\epsilon^{-1}))<\epsilon$, and we have a smooth function $u$ on $B(o,1)$ satisfying $\sup_{B(o,1)}|u|<\epsilon$ and

[TABLE]

Then we can find a constant $\beta$, an automorphism $g$ of $\mathbf{C}^{n}$ fixing the origin $o$, and a smooth function $u^{\prime}$ on $B(o,1)$ satisfying

1. (1)

$\beta g^{*}(\omega+\sqrt{-1}\partial\overline{\partial}u)=\omega+\sqrt{-1}\partial\overline{\partial}u^{\prime}$, 2. (2)

$(\beta g^{*}\omega)^{n}=\omega^{n}$, 3. (3)

$\sup_{B(o,\lambda)}|u^{\prime}|\leq\lambda^{2+\alpha}\sup_{B(o,1)}|u|$.

In this work in the property (3) the constant $\lambda^{2}$ would suffice, however we expect that the result has other applications that need the better constant $\lambda^{2+\alpha}$. The strategy of the proof is to show that once $\epsilon$ is sufficiently small, the function $u$ is close to a harmonic function on $C(Y)$. We then modify $u$ by subtracting a pluriharmonic function, and applying an automorphism to get a function $u^{\prime}$ with faster than quadratic decay using Lemma 4. The main difficulty is to control the behavior of $u$ near the singular set of $C(Y)$. We will achieve this by using the maximum principle with a suitable barrier function in order to show that if $u$ concentrates near the singular set, then $u$ decays rapidly when passing from $B(o,1)$ to $B(o,1/2)$. Note that much of the argument works in more general settings than what we are considering. The one place where we will use the fairly explicit form of the reference metric $\omega_{0}$ on $\mathbf{C}^{n}$ is when we need to control the action of the automorphisms on the Kähler potential of $\omega$ near the singular set, using Proposition 2.

To begin, we note that on any compact set away from the singular set of $C(Y)$ we can apply the small perturbation result of Savin [26] to get regularity of $|u|$ once $\epsilon$ is sufficiently small. For $\theta>0$ we define the set $N_{\theta}$ to be the $\theta$-neighborhood of the singular set under the Gromov-Hausdorff approximation:

[TABLE]

where we note that on $C(Y)$ the function $2|x|^{2\frac{n-2}{n-1}}$ is the distance squared from the singular set under the Ricci flat cone metric.

Lemma 8**.**

Let $\theta>0$. There exist $C_{k}>0$ depending on $\theta$ with the following property. If, in the setting of Proposition 7, $\epsilon$ is sufficiently small (depending on $\theta$), then on $B(o,{1-\theta})\setminus N_{\theta}$ we have

[TABLE]

Proof.

Note that by the Cheeger-Colding theory [4] and Anderson’s epsilon regularity result [1], we can bound the harmonic radius of $\omega$ on $B(o,1-\theta)\setminus N_{\theta}$ once $\epsilon$ is sufficiently small. Then for any $\delta>0$, if $\sup_{B(o,1)}|u|$ is sufficiently small, then we can apply Savin’s result [26] in harmonic coordinates to obtain $|u|_{C^{3}(B(o,1-\theta/2))}<\delta$. We have the equation

[TABLE]

which we can write as

[TABLE]

If $\delta$ is sufficiently small (so that $\sqrt{-1}\partial\overline{\partial}u$ is small), then we can view this as a uniformly elliptic homogeneous equation for $u$ with $C^{\alpha}$ coefficients (the expression in square brackets determining the coefficients of the equation). The Schauder estimates imply that on $B(o,1-3\theta/4)$ we have the bound $|u|_{C^{2,\alpha}}\leq C_{2}\sup_{B(o,1)}|u|$. We can bootstrap this estimate to obtain the required higher order bounds. ∎

We will apply this result in the following form several times.

Lemma 9**.**

Suppose that we have a sequence of functions $u_{i}$ as in Proposition 7 such that $\epsilon_{i}\to 0$. Let $v_{i}=(\sup_{B(o_{i},1)}|u_{i}|)^{-1}u_{i}$. Then there is a (possibly vanishing) bounded harmonic function $h$ on $B(0,1)\subset C(Y)$ such that under the Gromov-Hausdorff convergence, for a subsequence, we have $v_{i}\to h$ in $C^{\infty}$ uniformly on $B(o_{i},1-\theta)\setminus N_{\theta}$ for any $\theta>0$.

Proof.

By Lemma 8, for any given $\theta>0$ the functions $v_{i}$ satisfy uniform $C^{\infty}$ bounds on $B(o_{i},1-\theta)\setminus N_{\theta}$ for sufficiently large $i$, and they are all bounded by 1 on $B(o_{i},1)$. By a diagonal argument we can find a subsequence such that $v_{i}\to h$ for a function $h$ on $B(0,1)\subset C(Y)$, with the convergence taking place in $C^{\infty}$ on any $B(o_{i},1-\theta)\setminus N_{\theta}$.

To see that $h$ is harmonic, note that just like the equation (4.6), $v_{i}$ satisfies

[TABLE]

By Lemma 8, on $B(o_{i},1-\theta)\setminus N_{\theta}$ we have $u_{i}\to 0$ in $C^{\infty}$, and so passing to the limit in these equations we get that $h$ is harmonic on $B(0,1-\theta)\setminus N_{\theta}$. This holds for any $\theta$, and so $h$ is harmonic on the regular part of $B(0,1)$. In addition $h$ is bounded, so it is harmonic in a weak sense across the singular set too. ∎

4.1. Construction of a barrier function

We suppose that we are in the setting of Proposition 7. The following provides the barrier function used in the maximum principle argument below.

Proposition 10**.**

There is a constant $D>0$ with the following property. Let $\theta>0$. There is a constant $C_{\theta}>0$ such that if $\epsilon$ is sufficiently small (depending on $\theta$), then there is a smooth real function $v$ on $B(o,1)$ satisfying the following properties:

1. (1)

$|\partial\bar{\partial}v|_{\omega}<C_{\theta}$* on $B(o,1-\theta/2)$,* 2. (2)

$v(q)>D^{-1}\theta^{-1/2}$* whenever $q\in N_{\theta}\cap\partial B(o,1-\theta)$.* 3. (3)

$v>D^{-1}$* on $B(o,1)$, and $v<D$ on $B(o,1/2)$.* 4. (4)

On $B(o,1-\theta/2)$ the function $v$ satisfies the differential inequality

[TABLE]

where the $\mu_{i}$ are the eigenvalues of $\sqrt{-1}\partial\overline{\partial}v$ relative to $\omega$, and $\mu_{\max}$ is the largest eigenvalue.

Proof.

Let $(z,0)$ be a point in the singular set of $C(Y)$, with $|z|=1$, and let $q\in B(o,2)$ be within $\epsilon$ of $(z,0)$ under the Gromov-Hausdorff approximation. Using that $C(Y)$ is a cone also when centered at $(z,0)$, once $\epsilon$ is sufficiently small we can apply [23, Proposition 3.1] to find a good Kähler potential $\varphi$ for $\omega$ on $B(q,3)$, in the sense that we have $\omega=\sqrt{-1}\partial\overline{\partial}\varphi$, while also

[TABLE]

on $B(q,3)$. By adding a constant of order $\Psi(\epsilon)$ we can assume that $\varphi>0$. We have $\Delta\varphi=n$ (using the complex Laplacian), and from the Cheeger-Colding estimate [3] together with the Cheng-Yau gradient estimate [5] we have

[TABLE]

where we emphasize that we are taking the $(1,0)$-part of the derivative of $\varphi$. At the same time from the Bochner formula (computing in normal coordinates)

[TABLE]

since $\varphi_{i\bar{j}}=\omega_{i\bar{j}}$. So

[TABLE]

and the mean value inequality implies that

[TABLE]

Let us now consider the function $\varphi^{-3/4}$. We have

[TABLE]

Fix $y\in B(o,1)$. We can choose orthonormal coordinates for $\omega$ at $y$ such that $\partial\varphi=\varphi_{1}dz^{1}$, and $\sqrt{-1}\partial\overline{\partial}\varphi$ is the identity matrix. By the estimate above, we have $|\varphi_{1}|^{2}\leq\varphi+\Psi(\epsilon)$. The eigenvalues of $\sqrt{-1}\partial\overline{\partial}\varphi^{-3/4}$ therefore satisfy

[TABLE]

The maximum eigenvalue is necessarily $\mu_{1}$, and so

[TABLE]

As long as $\varphi>7\Psi(\epsilon)$, we obtain

[TABLE]

In particular by (4.9) this holds if $y\in B(o,1-\theta)$, and $\epsilon$ is sufficiently small (depending on $\theta$).

We also have $\varphi>\theta^{2}/4$ on $B(o,1-\theta)$ once $\epsilon$ is sufficiently small, and so from the bounds for the eigenvalues we have $|\sqrt{-1}\partial\overline{\partial}\varphi^{-3/4}|<C_{\theta}$ on $B(o,1-\theta)$ (where $C_{\theta}$ is of order $\theta^{-7/2}$, although we do not need this). On $B(o,1/2)$ we have $\varphi>1/10$ once $\epsilon$ is sufficiently small, and this leads to an upper bound $\varphi^{-3/4}<10^{3/4}$ on $B(o,1/2)$. At the same time $\varphi<1$ on $B(o,1)$ for sufficiently small $\epsilon$, and so $\varphi^{-3/4}>1$ on $B(o,1)$.

Note that once $\epsilon$ is sufficiently small, we have $\varphi<16\theta^{2}$ on $B(q,4\theta)$, and so

[TABLE]

This means $\varphi^{-3/4}$ is large near $q$, but we want a function that is large at all points in $N_{\theta}\cap\partial B(o,1-\theta)$. To achieve this, we define $v$ to be an average of functions $\varphi^{-3/4}$ constructed for different points in $S^{1}\times\{0\}\in\mathbf{C}\times A_{1}$. For a given $\theta>0$, we pick $z_{1},\ldots,z_{K}$ on the unit circle, and $q_{i}\in B(o,2)$ which are $\epsilon$-close to $(z_{i},0)\in C(Y)$ under our Gromov-Hausdorff approximation, so that the $4\theta$-balls $B(q_{i},4\theta)$ cover $N_{\theta}\cap\partial B(o,1-\theta)$. For sufficiently small $\epsilon$ we can achieve this with $K<c\theta^{-1}$, for a uniform $c$. We consider the functions $\varphi_{i}^{-3/4}$ constructed as above, based at the points $q_{i}$, and define

[TABLE]

Then $v$ satisfies the required properties:

1. (1)

$|\sqrt{-1}\partial\overline{\partial}v|_{\omega}<C_{\theta}$, since we are taking an average of functions that satisfy this estimate. 2. (2)

If $q\in N_{\theta}\cap B(o,1-\theta)$, then by assumption, there is a $q_{i}$ such that $q\in B(q_{i},4\theta)$. It follows that

[TABLE]

which gives the required lower bound. 3. (3)

We have $\varphi_{i}^{-3/4}>1$ on $B(o,1)$ for all $i$, so $v$ satisfies the same estimate. Similarly, $\varphi_{i}^{-3/4}<10^{3/4}$ on $B(o,1/2)$, and so $v$ satisfies the same. 4. (4)

Each $\varphi_{i}^{-3/4}$ satisfies the required differential inequality (4.17) on $B(o,1-\theta/2)$, and the expression $\sum_{i}\mu_{i}+\mu_{max}$ is convex on the space of Hermitian matrices. Therefore $v$ satisfies the same differential inequality.

∎

Note that this result can easily be generalized to other cones of the form $C(Y)=\mathbf{C}^{k}\times C(Y^{\prime})$, where $Y^{\prime}$ has an isolated singularity, but we have crucially used that all singular points in $C(Y)$ can be taken to be a vertex of $C(Y)$. We still expect that with some additional work a similar barrier function can be constructed for more general cones.

We now use the maximum principle to obtain the following important decay property.

Proposition 11**.**

There is a constant $C>0$ with the following property. Let $A>10$. There exists $\theta>0$ depending on $A$, such that if in the setting of Proposition 7 $\epsilon$ is sufficiently small (depending on $A,\theta$), and

[TABLE]

then

[TABLE]

Note that $C$ does not depend on $A,\theta$.

We first need the following lemma.

Lemma 12**.**

Suppose that $\mu_{i}>-1$ are constants for $i=1,\ldots,n$ such that

[TABLE]

There is a $\delta_{0}>0$ depending only on $n$, such that if $\mu_{i}<\delta_{0}$ for all $i$, then

[TABLE]

while if $\mu_{i}>-\delta_{0}$ for all $i$, then

[TABLE]

Here $\mu_{max},\mu_{min}$ are the largest and smallest of the $\mu_{i}$.

Proof.

Suppose that $\mu_{max}=\delta<\delta_{0}$. For all $j$ we have

[TABLE]

We can choose $\delta_{0}$ so that if $0<\delta<\delta_{0}$, then

[TABLE]

so we find $\mu_{j}\geq-n\delta$ for all $j$. Then if $\delta<1$, we have

[TABLE]

for a constant $C_{n}$ depending only on $n$, since the remaining terms are all at least quadratic in the $\mu_{i}$. It follows that

[TABLE]

Finally, if $\delta$ is sufficiently small, then $C_{n}\delta^{2}\leq\delta$. The argument for the second statement is completely analogous. ∎

Proof of Proposition 11.

Let us choose $\theta=A^{-2}$, and assume $\epsilon$ is sufficiently small to apply Proposition 10. Let $\Lambda>0$ satisfy

[TABLE]

and suppose that

[TABLE]

In terms of $v$ given by Proposition 10, set $\tilde{v}=\Lambda^{-1}v$, so that $\tilde{v}>u$ on $\partial B(o,1-\theta)$ by properties (2) and (3). We assume in addition that $\epsilon$ is sufficiently small (and so $\sup_{B(o,1)}|u|$ is sufficiently small) so that $\Lambda^{-1}C_{\theta}<\delta_{0}$ for the $\delta_{0}$ from Lemma 12.

Claim. $\tilde{v}>u$ on $B(o,1-\theta)$.

Proof of Claim. If this were not the case, then setting

[TABLE]

the graph of $\tilde{v}+t_{0}$ will lie above the graph of $u$, and the two graphs will touch at a point $q\in B(o,1-\theta)$. At $q$ we must have $\sqrt{-1}\partial\overline{\partial}u(q)\leq\sqrt{-1}\partial\overline{\partial}\tilde{v}(q)$. In orthonormal coordinates at $q$ we have $\sqrt{-1}\partial\overline{\partial}\tilde{v}(q)\leq\Lambda^{-1}C_{\theta}\mathrm{Id}$ by property (1) in Proposition 10, and so the eigenvalues $\mu_{i}$ of $\sqrt{-1}\partial\overline{\partial}u(q)$ are bounded above by $\Lambda^{-1}C_{\theta}$. Since $\Lambda^{-1}C_{\theta}<\delta_{0}$, Lemma 12 implies that $\sum\mu_{i}+\mu_{max}\geq 0$, but this contradicts property (4) in Proposition 10, since $\sqrt{-1}\partial\overline{\partial}u(q)\leq\sqrt{-1}\partial\overline{\partial}\tilde{v}(q)$. This proves the claim.

Using the claim, it follows from property (3) that

[TABLE]

which gives the required upper bound for $u$.

The lower bound for $u$ is proved similarly, just comparing with the function $-\Lambda^{-1}v$ instead, and using the second statement in Lemma 12. ∎

4.2. Proof of Proposition 7

We prove Proposition 7 by contradiction. Let us suppose that we have a sequence $u_{i}$ as in the statement of the proposition, on balls $B(o_{i},1)$, with corresponding constants $\epsilon_{i}\to 0$, such that the conclusion of the proposition fails. We will show that along a subsequence, for sufficiently large $i$ we can find $\beta_{i},g_{i},u^{\prime}_{i}$ satisfying the required properties, giving a contradiction.

Step 1. Let us write $\kappa_{i}=\sup_{B(o_{i},1)}|u_{i}|\to 0$. By Lemma 9 we have a harmonic function $h$ on $B(0,1)\in C(Y)$ such that, after choosing a subsequence, $\kappa_{i}^{-1}u_{i}\to h$ in $C^{\infty}$ uniformly on $B(o_{i},1-\theta)\setminus N_{\theta}$ for any $\theta>0$. Note that we have $|h|\leq 1$, and it is possible that $h=0$. Let us write $h=h^{\leq 2}+h^{>2}$ for the decomposition of $h$ into pieces with at most quadratic and faster than quadratic growth. In addition we decompose $h^{\leq 2}=h_{ph}+h_{aut}$, where $h_{ph}$ is pluriharmonic, and $h_{aut}$ is in the span of the functions of type (2) and (3) in Lemma 3. From Lemma 4 we obtain a real holomorphic vector field $V$ on $\mathbf{C}^{n+1}$ preserving the hypersurfaces

[TABLE]

and a constant $\beta$, such that $L_{V}\Omega=n\beta\Omega$, and at the same time on $C(Y)\subset\mathbf{C}^{4}$ we have

[TABLE]

We define the automorphism $g_{i}=\exp(\kappa_{i}V)$ on $\mathbf{C}^{n+1}$, and the constants $\beta_{i}=\kappa_{i}\beta$.

For any $\theta>0$, the spaces $B(o_{i},1)\setminus N_{\theta}$ converge smoothly to $B(0,1)\setminus N_{\theta}$ inside $\mathbf{C}^{n+1}$, and for sufficiently large $i$ we can use the nearest point projection to identify them. On $B(o_{i},1)$ we have Kähler potentials $\varphi_{i}$ for $\omega_{i}$, which converge smoothly to $|z|^{2}+|x|^{2\frac{n-1}{n-2}}$ as $i\to\infty$, uniformly on $B(o_{i},1)\setminus N_{\theta}$ for any $\theta>0$. It follows from this, that

[TABLE]

and so we also have

[TABLE]

Here $\Psi(i^{-1}\,|\,\theta)$ denotes a function, which for fixed $\theta$ converges to zero as $i\to\infty$. At the same time, using Proposition 2, we have a uniform bound $|V|_{\omega_{i}}<C$ on $B(o_{i},1)$, since this ball is a ball centered at the origin in our reference metric $(\mathbf{C}^{n},\omega_{0})$ scaled down to unit size. Together with the uniform gradient bound for $\varphi_{i}$ (since $\Delta_{\omega_{i}}\varphi_{i}=n$), this implies

[TABLE]

To deal with $h_{ph}$ note that $h_{ph}$ is in the span of the real and imaginary parts of $1,z,z^{2},x_{i}$, and so $h_{ph}$ also defines a pluriharmonic function $h_{ph,i}$ on $B(o_{i},1)$ for all $i$ under the embeddings into $\mathbf{C}^{n+1}$. For any $\theta>0$ we have

[TABLE]

where as above, we are using the nearest point projection on $B(o_{i},1)\setminus N_{\theta}$ for sufficiently large $i$ to view $h_{ph}$ as a function on $B(o_{i},1)$. We also clearly have

[TABLE]

We can now define

[TABLE]

By the construction we have

[TABLE]

and $e^{-\beta_{i}}g_{i}^{*}\omega_{i}$ has the same volume form as $\omega_{i}$. By the estimates above, we have

[TABLE]

and also

[TABLE]

Letting $\theta\to 0$, we find that

[TABLE]

where near the singular set we use a Gromov-Hausdorff approximation to view $h_{i}^{>2}$ as a function on $B(o_{i},1)$.

Since $h_{i}^{>2}$ has faster than quadratic growth, there exists an $\alpha>0$ (depending only on the cone $C(Y)$), such that

[TABLE]

Here, and below, for any ball $B$ we define

[TABLE]

to be the $L^{2}$-norm normalized by the volume of the ball $B$.

We therefore have

[TABLE]

while also

[TABLE]

for a constant $C_{\lambda}$ depending on $\lambda$. Combining these, we get

[TABLE]

Once $i$ is sufficiently large (depending on $\lambda$), we get

[TABLE]

Step 2. Let us apply the construction in Step 1 to $8\lambda$ instead of $\lambda$, and let us scale the balls $B(o_{i},8\lambda)$ up to unit size. We denote the origins of the scaled up balls by $o_{i}^{\prime}$, and also let $\omega_{i}^{\prime}=(8\lambda)^{-2}\beta_{i}g_{i}^{*}\omega_{i}$. Letting $U_{i}^{\prime}=(8\lambda)^{-2}u_{i}^{\prime}$, on $B(o_{i}^{\prime},1)$ we have

[TABLE]

and for fixed $\lambda$, as $i\to\infty$, the metrics $\omega^{\prime}_{i}$ on $\mathbf{C}^{n}$ satisfy the same assumptions as $\omega$ in the statement of Proposition 7 for arbitrarily small $\epsilon$. By Step 1, (replacing $C$ by a larger constant if necessary) we have

[TABLE]

By Lemma 9 we have a harmonic function $H$ on $B(0,1)\subset C(Y)$, such that after choosing a subsequence

[TABLE]

the convergence being in $C^{\infty}$ uniformly on $B(o_{i}^{\prime},3/4)\setminus N_{\theta}$ for any $\theta>0$. There are two cases:

• •

Suppose that $H\not=0$. Then by the $L^{\infty}$ bound for harmonic functions on $C(Y)$, we have

[TABLE]

It follows that for any $\theta>0$, once $i$ is sufficiently large, we have

[TABLE]

and so using (4.53) we have

[TABLE]

• •

Suppose that $H=0$. Then for any $\theta>0$ we have

[TABLE]

once $i$ is sufficiently large.

In either case, applying Proposition 11 to $B(o_{i}^{\prime},1/2)$, we can choose $\theta>0$ depending on $\lambda$, such that for sufficiently large $i$ we have

[TABLE]

for a constant $C^{\prime}$ depending only on $C(Y)$. After rescaling, we get the bound $\sup_{B(o_{i},2\lambda)}|u_{i}^{\prime}|\leq 64C^{\prime}\lambda^{2+2\alpha}\kappa_{i}$, and from (4.42) we have

[TABLE]

Letting $u_{i}^{\prime\prime}=e^{\beta_{i}}(g_{i}^{-1})^{*}u_{i}^{\prime}$ we have

[TABLE]

as long as $g_{i}^{-1}(B(o_{i},\lambda))\subset B(o_{i},2\lambda)$, and $e^{\beta_{i}}<3/2$. Both of these estimates will hold once $i$ is sufficiently large (depending on $\lambda$). Finally we just need to ensure that $\lambda$ is sufficiently small so that $100C^{\prime}\lambda^{2+2\alpha}<\lambda^{2+\alpha}$.

4.3. Proof of Theorem 1

We now prove the main result, Theorem 1. Suppose that we have a Calabi-Yau metric $(X^{\prime},\eta)$ with $X^{\prime}$ biholomorphic to $\mathbf{C}^{n}$, and with tangent cone $\mathbf{C}\times A_{1}$ at infinity. Let us write $(X,\omega_{0})$ for our reference metric on $\mathbf{C}^{n}$ discussed in Section 2. We have the origin $o\in X$, and using Lemma 6 we also have a distinguished basepoint $o^{\prime}\in X^{\prime}$. By the discussion in Section 2 we have embeddings $F_{i}:X\to\mathbf{C}^{n+1}$ such that $F_{i}(o)=0$, and the image $F_{i}(X)$ has equation

[TABLE]

where $a_{i}=2^{-in/(n-2)}$. At the same time using Proposition 5 we have embeddings $F_{i}^{\prime}:X^{\prime}\to\mathbf{C}^{n+1}$ such that $F_{i}^{\prime}(o)=0$, and the image $F_{i}^{\prime}(X^{\prime})$ satisfies the equation

[TABLE]

with $a_{i}^{\prime}>0$. In addition the unit balls for the scaled down metrics $2^{-2i}\omega_{0},2^{-2i}\eta$ converge in the Gromov-Hausdorff sense to the unit ball $B(0,1)\subset C(Y)$, and the maps $F_{i},F_{i}^{\prime}$ converge to the standard embedding $B(0,1)\subset\mathbf{C}^{n+1}$ as $i\to\infty$.

We need to find suitable $j(i)$ such that after a further scaling by a bounded factor, the images $F_{j(i)}^{\prime}(X^{\prime})$ satisfy the same equations as $F_{i}(X)$. Since $a_{i}/a_{i+1}=2^{n/(n-2)}$ and $a_{i}^{\prime}/a_{i+1}^{\prime}\to 2^{n/(n-2)}$, for all sufficiently large $i$ we can find $j(i)$ such that

[TABLE]

for a dimensional constant $C_{n}$, and $j(i)\to\infty$ as $i\to\infty$. Let us compose $F_{j(i)}^{\prime}$ with an automorphism $g_{i}$ of $\mathbf{C}^{n+1}$ of the form $(z,\mathbf{x})\mapsto(c_{i}z,c_{i}^{\frac{n-1}{n-2}}\mathbf{x})$ for $c_{i}>0$. Then $g_{i}\circ F_{j(i)}^{\prime}(X^{\prime})$ satisfies the equation

[TABLE]

so we can choose $c_{i}\in(C_{n}^{-1},C_{n})$ so that $g_{i}\circ F_{j(i)}^{\prime}(X^{\prime})$ satisfies the same equation as $F_{i}(X)$. The balls $B_{\eta}(o^{\prime},c_{i}^{-1}2^{-j(i)})$ with the scaled metrics $\eta_{i}=c_{i}^{-2}2^{-2j(i)}\eta$ still converge to the unit ball $B(0,1)\subset C(Y)$, and on these balls the maps $g_{i}\circ F^{\prime}_{j(i)}(X^{\prime})$ still converge to the standard embedding of $B(0,1)$ into $\mathbf{C}^{n+1}$ as $i\to\infty$. Moreover the volume form of $\eta_{i}$ is the pullback of $\sqrt{-1}^{n^{2}}\Omega\wedge\bar{\Omega}$ under $g_{i}\circ F_{j(i)}^{\prime}$.

Let us write $\omega_{i}=2^{-2i}\omega_{0}$, and $B_{i}$ for the unit ball around $o$ with the metric $\omega_{i}$. Similarly let $B_{i}^{\prime}$ be the unit ball around $o^{\prime}$ with the metric $\eta_{i}$. We have biholomorphisms $\Phi_{i}:X\to X^{\prime}$ defined by

[TABLE]

which satisfy $\Phi_{i}(o)=o^{\prime}$, and $\Phi_{i}^{*}(\eta^{n})=\omega_{0}^{n}$. We claim that on the ball $B_{i}$ we have

[TABLE]

For this, let $x\in B_{i}$, and $x^{\prime}=\Phi_{i}(x)$. By the construction and Proposition 5 we have $\Psi(i^{-1})$-Gromov-Hausdorff approximations $G:B_{i}\to B(0,1)$ and $G^{\prime}:B_{i}^{\prime}\to B(0,1)$ satisfying $G(o)=0,G^{\prime}(o^{\prime})=0$ such that viewing $B(0,1)\subset\mathbf{C}^{n+1}$ under the standard embedding, we have $|G(x)-F_{i}(x)|<\Psi(i^{-1})$ and $|G^{\prime}(x^{\prime})-g_{i}\circ F_{j(i)}^{\prime}(x^{\prime})|<\Psi(i^{-1})$. Note that $F_{i}(x)=g_{i}\circ F_{j(i)}^{\prime}(x^{\prime})$ by assumption, and so $|G(x)-G^{\prime}(x^{\prime})|<\Psi(i^{-1})$. At the same time, under the cone metric on $B(0,1)$, the distance from [math] is Hölder continuous with respect to the Euclidean distance (it is given up to a factor by $|x|^{\frac{n-2}{n-1}}$), so this means

[TABLE]

Since $G,G^{\prime}$ are Gromov-Hausdorff approximations, we get

[TABLE]

as claimed.

The balls $B_{i},B_{i}^{\prime}$ are both $\Psi(i^{-1})$- Gromov-Hausdorff close to the unit ball in $C(Y)$. For sufficiently large $i$ we can then use [23, Proposition 3.1] to find Kähler potentials $\varphi_{i},\varphi_{i}^{\prime}$ for $\omega_{i},\eta_{i}$, such that

[TABLE]

Using (4.67) this means that on $B_{\omega_{i}}(o,1)$ we can write

[TABLE]

with $\sup_{B_{\omega_{i}}(o,1)}|u_{i}|<\Psi(i^{-1})$.

We will next apply Proposition 7. We can assume that the $\lambda$ in the proposition is of the form $\lambda=2^{-m}$ for an integer $m$. We can also choose $i_{0}>0$, such that the assumptions of Proposition 7 hold for $\omega_{i}$ and $u_{i}$ on $B_{\omega_{i}}(o,1)$, for all $i\geq i_{0}$. Let us fix a large $k>0$, and apply the proposition for $i=i_{0}+km$. We have

[TABLE]

where $\lim_{k\to\infty}\epsilon_{k}=0$. We find a $\beta_{k},g_{k}$ and $u_{k}^{\prime}$ such that

[TABLE]

on $B_{\omega_{i_{0}+km}}(o,1)$ together with the estimate

[TABLE]

Note that $B_{\omega_{i_{0}+km}}(o,\lambda)=B_{\omega_{i_{0}+(k-1)m}}(o,1)$. Scaling (4.73) up by a factor of $\lambda^{-2}$, we have

[TABLE]

and $\sup_{B_{\omega_{i_{0}+(k-1)m}}(o,1)}|\lambda^{-2}u_{k}^{\prime}|\leq\lambda^{\alpha}\epsilon_{k}\leq\epsilon_{k}$, where we dropped the $\lambda^{\alpha}$ factor. Note that by construction the volume forms of $\lambda^{-2}\beta_{k}^{*}g_{k}^{*}\Phi_{k}^{*}\eta_{i_{0}+km}$ and $\omega_{i_{0}+(k-1)m}$ are equal, and so we can apply Proposition 7 again, iterating the above argument. After $k$ steps we obtain a constant $\Lambda_{k}$, a biholomorphism $G_{k}:X\to X^{\prime}$ satisfying $G_{k}(o)=o^{\prime}$, and a function $U_{k}$ such that

[TABLE]

on $B_{\omega_{i_{0}}}(o,1)$, together with the estimate

[TABLE]

such that

[TABLE]

Here we have absorbed the additional scaling between $\eta$ and $\eta_{k}$ into the constant $\Lambda_{k}$. Note that fixing $i_{0}$ we can take $k\to\infty$, and once $\epsilon_{k}$ is sufficiently small, we can apply Savin’s small perturbation result [26] to find that on $B_{\omega_{i_{0}}}(o,1/2)$ we have $U_{k}\to 0$ in $C^{\infty}$. Since $G_{k}(o)=o^{\prime}$, we find that for sufficiently large $k$

[TABLE]

and moreover $G_{k}^{*}(\Lambda_{k}\eta)\to\omega_{i_{0}}$ in $C^{\infty}$ on $B_{\omega_{i_{0}}}(o,1/2)$. If $\Lambda_{k}\to 0$, then this is a contradiction, since $\eta$ is not flat, and so the curvature of $B_{\Lambda_{k}\eta}(o^{\prime},1/4)$ blows up as $\Lambda_{k}\to 0$. Similarly $\Lambda_{k}\to\infty$ leads to a contradiction since $\omega_{0}$ is not flat. Choosing a subsequence we can assume $\Lambda_{k}\to\Lambda_{\infty}>0$. It follows that we can then take a limit $G_{k}\to G_{\infty}$ on $B_{\omega_{i_{0}}(o,1/4)}$ which gives a holomorphic isometry

[TABLE]

We can repeat the same argument for any $i>i_{0}$, and extract a global holomorphic isometry between $(X,\omega_{0})$ and $(X^{\prime},\Lambda\eta)$ for a suitable $\Lambda$.

5. Further directions

The approach that we used to prove Theorem 1 can be applied in more general situations. One natural generalization would be to study the uniqueness of all of the metrics constructed in the author’s work [30], or by Conlon-Rochon [12], given their tangent cones. The places where we used the specific choice $\mathbf{C}\times A_{1}$ for the tangent cone were in Lemma 4 in order to understand the quadratic growth harmonic functions, and in Proposition 5 which allowed us to construct embeddings of a given Calabi-Yau space as a specific hypersurface. When we consider more general tangent cones, then these results need to be suitably modified. We expect that in general the Calabi-Yau metric with a given tangent cone is not unique, however we hope that our methods can be used to describe the moduli space of such metrics.

To illustrate this, let us consider the next simplest example, namely the metric $\omega_{0}$ on $\mathbf{C}^{3}$ with tangent cone $\mathbf{C}\times A_{2}$ at infinity, constructed by viewing $\mathbf{C}^{3}\subset\mathbf{C}^{4}$ as the hypersurface

[TABLE]

This metric has the property that we have holomorphic functions $z,x_{1},x_{2},y$ whose degrees satisfy $d(z)=1,d(x_{1})=3,d(x_{2})=3,d(y)=2$, and which satisfy (5.1). Suppose now that $(X,\eta)$ is another Calabi-Yau metric with the same tangent cone, and we try to argue as in Proposition 5. The same arguments show that we can embed $X$ into $\mathbf{C}^{4}$ as a hypersurface given by a linear equation in monomials of total degree at most 6. Moreover this equation is a small perturbation of the equation $x_{1}^{2}+x_{2}^{2}+y^{3}=0$ defining the tangent cone. As in Proposition 5 we can perform simplifications, but we cannot always reduce to the equation (5.1). Instead we may end up with an equation of the form

[TABLE]

for small constants $a,b$, with $a\neq 0$. Note that this defines a hypersurface biholomorphic to $\mathbf{C}^{3}$. When $b\neq 0$, then we cannot make the change of coordinate $z^{\prime}=z+a^{-1}by$ to reduce to an equation of the form (5.1), since $y$ has faster growth than $z$. Indeed, we expect that one can construct a one-parameter family of inequivalent Calabi-Yau metrics on $\mathbf{C}^{3}$ with tangent cone $\mathbf{C}\times A_{2}$, using the methods from [20, 12, 30], viewing $\mathbf{C}^{3}\subset\mathbf{C}^{4}$ as the hypersurface

[TABLE]

More precisely we expect the following.

Conjecture 13**.**

Up to scaling and isometry there is a one parameter family of Calabi-Yau metrics on $\mathbf{C}^{3}$ with tangent cone $\mathbf{C}\times A_{2}$ at infinity.

In view of the gluing construction by Li [19] of collapsing Calabi-Yau metrics on threefolds (see the discussion in Section 4.2), such metrics could arise as a suitable blowup limit of a collapsing family of CY metrics on a threefold that has a fibration locally of the form $(x_{1},x_{2},y)\mapsto x_{1}^{2}+x_{2}^{2}+y^{3}$.

A different generalization would be to consider Calabi-Yau metrics on more general spaces than $\mathbf{C}^{n}$. The crucial prerequisite for applying the Donaldson-Sun theory in Section 3, as well as [23, Proposition 3.1], was that the metric $\omega_{0}$ is $\partial\bar{\partial}$-exact, and we expect that our methods can be extended to classifying such exact Calabi-Yau metrics. For instance the smoothing $Q^{n}\subset\mathbf{C}^{n}$ of the $n$-dimensional $A_{1}$ singularity,

[TABLE]

is expected to admit a Calabi-Yau metric with tangent cone $\mathbf{C}\times A_{1}$ in terms of the $(n-1)$-dimensional $A_{1}$ singularity (see e.g. [19, Section 4.2]), and the methods used in Theorem 1 could lead to a uniqueness result for this metric.

More generally, from the argument in Proposition 5 we can read off which manifolds can admit a $\partial\bar{\partial}$-exact Calabi-Yau metric with a given tangent cone. For instance the following is a natural conjecture to make.

Conjecture 14**.**

Let $n>4$. The only $\partial\bar{\partial}$-exact Calabi-Yau manifolds of dimension $n$ with tangent cone $\mathbf{C}\times A_{1}$ are $\mathbf{C}\times Q^{n-1}$, $\mathbf{C}^{n}$ and $Q^{n}$. Moreover up to scaling and isometry each of these manifolds admits a unique such Calabi-Yau metric.

The three cases correspond to the function $\tilde{f}_{i}$ in the equation analogous to (3.12) having degree 0, 1 or 2. When $n\leq 4$ then there would be more possibilities. For both Conjectures 13 and 14 we expect that the proof of Theorem 1 can be extended to prove the classification results, once the corresponding existence results are shown using the techniques of [20, 12, 30].

Note that some of the results of Donaldson-Sun [15] can also be extended to the case when the metric is not exact, under the assumption that the tangent cone is smooth away from the vertex (see Liu [21]). This is closer to the setting of asymptotically conical Calabi-Yau metrics considered by Conlon-Hein [10] who also obtained classification results for Calabi-Yau metrics with prescribed tangent cone. At the moment there is little that we can say in this direction about general Calabi-Yau manifolds with tangent cones that have non-isolated singularities, beyond the result in [24] that each tangent cone is a normal affine variety.