Gromov–Hausdorff limits of Kähler manifolds with Ricci curvature bounded below

Abstract

We show that non-collapsed Gromov–Hausdorff limits of polarized Kähler manifolds, with Ricci curvature bounded below, are normal projective varieties, and the metric singularities of the limit space are precisely given by a countable union of analytic subvarieties. This extends a fundamental result of Donaldson–Sun, in which 2-sided Ricci curvature bounds were assumed. As a basic ingredient we show that, under lower Ricci curvature bounds, almost Euclidean balls in Kähler manifolds admit good holomorphic coordinates. Further applications are integral bounds for the scalar curvature on balls, and a rigidity theorem for Kähler manifolds with almost Euclidean volume growth.

Appendix

In the proof of Proposition 3.1 we used the estimate of Cheeger–Jiang–Naber [CJN] for the volumes of tubular neighborhoods of the singular set, in order to control the cutoff functions $\psi ^3_i$. In this appendix we first give an alternative argument, following the approach of Chen–Donaldson–Sun [CDS15a], which is independent of the results in [CJN]. First note that the cone $V = {\mathbb {R}}^{2n-2} \times W$ for a two-dimensional cone $(W,o')$ has singular set ${\mathbb {R}}^{2n-2} \times \{o'\}$, and so we can directly see the required estimate for the volumes of its tubular neighborhoods. Therefore Propositions 3.4 and 3.5 hold without appealing to [CJN].

We can now argue similarly to [CDS15a] to show that the singular sets in any cone (V, o) arising in Proposition 3.1 have good cutoff functions (even closer to what we do are Propositions 12, 13 and 14 of the arXiv version of [Sze16]). More precisely, suppose that $(M_i, L_i, \omega _i)$ is a sequence as in Proposition 3.1 such that $(M_i, p_i)$ converges to a cone (V, o) for some $p_i\in M_i$. Recall that the singular set is $\Sigma \subset V$, consisting of points $q\in V$ such that $\lim _{r\rightarrow 0} r^{-2n}\mathrm {vol}(B(q,r)) \le \omega _{2n}-\epsilon$, where $\epsilon$ is obtained from Theorem 1.5. We then have the following.

Proposition 5.1

For any compact set $K\subset V$ and $\kappa > 0$ there is a function $\chi$ on V, equal to 1 on a neighborhood of $K\cap \Sigma$, supported in the $\kappa$-neighborhood of $K\cap \Sigma$, and such that $\int _K |\nabla \chi |^2 < \kappa$.

Proof

Let us fix $K, \kappa$. Suppose that we have distance functions $d_i$ on $B(p_i,2R) \sqcup B(o,2R)$, realizing the Gromov–Hausdorff convergence, where R is large so that $K\subset B(o,R)$. For $q\in B(o,R)$, and $\rho \in (0,1)$, define

$$\begin{aligned} V(i, q, \rho ) = \rho ^{2-2n} \int _{U_i(q, \rho )} (S_i + 2n) \omega _i^n, \end{aligned}$$

where $U_i(q,\rho ) = \{x \in B(p_i,2)\,:\, d_i(x, q) < \rho \}$. Note that $S_i + 2n \ge 0$.

Let us denote by ${\mathcal {D}}\subset K\cap \Sigma$ the set of points which have a tangent cone splitting off ${\mathbb {R}}^{2n-2}$, and let $S_2 = \Sigma {\setminus } {\mathcal {D}}$. Proposition 3.5 implies that for any $x\in {\mathcal {D}}$ there exists a $\rho _x > 0$ such that $V(i,x,\rho _x) < A$ for a fixed constant A, for sufficiently large i. At the same time by Proposition 3.5 we also have a constant $c_0 > 0$ (depending on $n, v, d, \epsilon$) such that for any $x\in {\mathcal {D}}$ and $\delta > 0$ there is an $r_x < \delta$ such that $V(i,x,r_x) > c_0$ for sufficiently large i (here note that the two dimensional cones appearing in tangent cones of points in ${\mathcal {D}}$ have cone angles bounded strictly away from $2\pi$).

By Cheeger–Colding’s [CC97] the Hausdorff dimension of $S_2$ is at most $2n-4$, so for any small $\epsilon > 0$ we can cover $S_2\cap K$ with balls $B_\mu$ such that

$$\begin{aligned} \sum _\mu r_\mu ^{2n-3} < \epsilon . \end{aligned}$$

The set

$$\begin{aligned} J = (K\cap \Sigma ) {\setminus } \cup _\mu B_\mu \end{aligned}$$

is compact, $J\subset {\mathcal {D}}$, and so it is covered by the balls $B(x,\rho _x)$ with $x\in {\mathcal {D}}$. We choose a finite subcover corresponding to $x_1,\ldots , x_N$, and set

$$\begin{aligned} \begin{aligned} W&= \bigcup _{j=1}^N B(x_j, \rho _{x_j}) \subset V, \\ W_i&= \bigcup _{j=1}^N U_i(x_j, \rho _{x_j}) \subset M_i. \end{aligned} \end{aligned}$$

For sufficiently large i we then get an estimate

$$\begin{aligned} \int _{W_i} (S_i + 2n) \omega _i^n < C, \end{aligned}$$

(5.1)

where C depends on $\epsilon , N$, but not on i.

We claim that the compact set $J \subset {\mathcal {D}}$ has finite $(2n-2)$-dimensional Hausdorff measure. To prove this, recall that for any small $\delta > 0$ and all $x\in {\mathcal {D}}\cap J$ we have $r_x < \delta$ such that $V(i,x,r_x) > c_0$ for large i. By a Vitali type covering argument we can find a disjoint, finite sequence of balls $B(y_k, r_{y_k})$ in W, for $k=1,\ldots , N'$ such that $B(y_k, 5r_{y_k})$ cover all of J. It follows that

$$\begin{aligned} {\mathcal {H}}^{2n-2}_\delta (J) \le \sum _{k=1}^{N'} 5^{2n-2} r_{y_k}^{2n-2}. \end{aligned}$$

At the same time for each $y_k$, we have the estimate

$$\begin{aligned} r_{y_k}^{2-2n} \int _{U_i(y_k,r_{y_k})} (S_i + 2n) \omega _i^n > c_0, \end{aligned}$$

for sufficiently large i. Taking i even larger we can assume that the $U_i(y_k, r_{y_k})$ are disjoint, since they converge in the Gromov–Hausdorff sense to the disjoint balls $B(y_k, r_{y_k})$. Using (5.1) we therefore have

$$\begin{aligned} \sum _{k=1}^{N'} c_0 r_{y_k}^{2n-2} < C. \end{aligned}$$

Since $\delta$ was arbitrary (and C is independent of $\delta$), this implies that ${\mathcal {H}}^{2n-2}(J) \le C'$.

It follows that J has capacity zero, in the sense that for any $\kappa > 0$ we can find a cutoff function $\eta _1$ supported in the $\kappa$-neighborhood of J, such that $\Vert \nabla \eta _1\Vert _{L^2} \le \kappa /2$, and $\eta _1 = 1$ on a neighborhood U of J (see for instance [Bou, Lemma 2.2] or [EG92, Theorem 3, p. 154]). The set $(K\cap \Sigma ) {\setminus } U$ is compact, and so it is covered by finitely many of our balls $B_\mu$ from before. Because of this, as in [DS14], we can find a good cutoff function $\eta _2$, with $\Vert \nabla \eta _2\Vert _{L^2} \le \kappa /2$ (if $\epsilon$ at the beginning was sufficiently small), such that $\eta _2$ is supported in the $\kappa$-neighborhood of $(K\cap \Sigma ){\setminus } U$ and with $\eta _2 = 1$ on a neighborhood of $(K\cap \Sigma ){\setminus } U$. Then $\eta = 1 - (1-\eta _1)(1-\eta _2)$ gives the required cutoff function. $\square$

We next prove a result essentially contained in Cheeger–Colding–Minicozzi [CCM95], that we used in the proof of Proposition 4.1.

Proposition 5.2

Let (V, o) denote a tangent cone of a non-collapsed limit space of manifolds with Ricci curvature bounded from below. Suppose that there are k linearly independent harmonic functions $u^1,\ldots , u^k$ on V that are homogeneous of degree one. Then we have a splitting $V = {\mathbb {R}}^k \times Y$.

Proof

By assumption we have a sequence $B(p_i, 2)$ of balls in Riemannian manifolds with $\mathrm {Ric} > -i^{-1}$, such that $B(p_i,2)\rightarrow B(o,2)$ in the Gromov–Hausdorff sense. We will prove the following: for any $\delta > 0$, we can find an $r > 0$ and $\delta$-splitting maps $u_i : B(p_i, r) \rightarrow {\mathbb {R}}^k$ for sufficiently large i. Since V is a cone, after scaling up by $r^{-1}$ and taking a diagonal sequence, we find a sequence $B(p_i', 1)\rightarrow B(o, 1)$ such that each $B(p_i', 1)$ admits an $i^{-1}$-splitting map. From this it follows that B(o, 1/2) splits an isometric factor of ${\mathbb {R}}^k$. For this, see Cheeger–Colding [CC96], or Cheeger–Naber [CN15, Definition 1.20, Lemma 1.21].

Before we begin let us recall the notion of a $\delta$-splitting map. A map $u=(u^1, \ldots , u^k): B(p,r) \rightarrow {\mathbb {R}}^k$ is a $\delta$-splitting map, if it is harmonic, and satisfies

• (1) $|\nabla u| < 1 + \delta$,

• (2) ,

• (3) .

Consider again our sequence $B(p_i, 2) \rightarrow B(o,2)$. We can assume that

and since the $u^\alpha$ are homogeneous, this implies that for all r we have

We can find a sequence of harmonic functions $u_i^{\alpha }$ on $B(p_i, 2)$ such that under the Gromov–Hausdorff convergence we have $u_i^\alpha \rightarrow u^\alpha$ uniformly on each compact set, and moreover for any $0< r < 2$,

(5.2)

Let $f_i$ denote a harmonic function of the form $u^\alpha _i$ or $\frac{1}{\sqrt{2}} (u_i^\alpha \pm u_i^\beta )$ for $\alpha \not = \beta$. By the Bochner formula $\Delta |\nabla f_i|^2 \ge -\Psi (i^{-1}) |\nabla f_i|^2$, and so by the mean value inequality, for sufficiently large i we have

$$\begin{aligned} \sup _{B(p_i, 1.5)} |\nabla f_i|^2 \le C. \end{aligned}$$

It follows that for any $0< r < 1.5$ we have

(5.3)

As the gradient of $f_i$ is uniformly bounded, we find the above convergence is uniform on the interval $a<r<1$, where $a>0$ is any constant.

Note that $\sup _{B(p_i, r)} |\nabla f_i|^2 \ge 1/2$, and so for large i

$$\begin{aligned} \sup _{B(p_i,1)} |\nabla f_i|^2 \le 2C \sup _{B(p_i,r)} |\nabla f_i|^2. \end{aligned}$$

Given $\epsilon > 0$, we can then choose $r_0 > 0$ depending on $\epsilon , C$, such that for all sufficiently large i there is some $r \in (r_0, 1/10)$, perhaps depending on i, satisfying

$$\begin{aligned} \sup _{B(p_i, 3r)} |\nabla f_i|^2 \le (1-\epsilon )^{-1} \sup _{B(p_i, r)} |\nabla f_i|^2. \end{aligned}$$

Consider now the functions $v_i = \sup _{B(p_i, 3r)}|\nabla f_i|^2 - |\nabla f_i|^2$. Then on $B(p_i, 3r)$, $v_i \ge 0$,

$$\begin{aligned} \Delta v_i \le \Psi (i^{-1}) \sup _{B(p_i, 3r)} |\nabla f_i|^2= \Psi (i^{-1}), \end{aligned}$$

and

$$\begin{aligned} \inf _{B(p_i, r)} v_i = \sup _{B(p_i, 3r)}|\nabla f_i|^2 - \sup _{B(p_i, r)} |\nabla f_i|^2 \le \epsilon \sup _{B(p_i, 3r)} |\nabla f_i|^2. \end{aligned}$$

From the weak Harnack inequality, once i is sufficiently large,

This implies

where C depends only on the non-collapsing constant, through the Sobolev inequality. Recall r (depending on i) has a lower bound $r_0$, and so by (5.3),

It follows that for sufficiently large i we have

$$\begin{aligned} \sup _{B(p_i, 2r_0)}|\nabla f_i|^2\le \sup _{B(p_i, 3r)}|\nabla f_i|^2 \le 1+C\epsilon . \end{aligned}$$

Therefore,

We now apply this to $f_i = u^\alpha _i$, and to $f_i = \frac{1}{\sqrt{2}}(u_i^\alpha \pm u_i^\beta )$, and use the polarization identity

$$\begin{aligned} \frac{1}{2} |\nabla (u_i^\alpha + u_i^\beta )|^2 - \frac{1}{2} |\nabla (u_i^\alpha - u_i^\beta )|^2 = 2 \langle \nabla u_i^\alpha , \nabla u_i^\beta \rangle . \end{aligned}$$

Using also (5.2) we find that for sufficiently large i (depending on $\epsilon$), we have

Since at the same time $|\nabla u_i^\alpha |^2 \le 1 + \Psi (\epsilon )$, we can use the Bochner formula, using a cutoff function $\phi$ as in Cheeger–Colding [CC96] supported in $B(p_i, 2r_0)$, equal to 1 in $B(p_i, r_0)$. We find that for sufficiently large i,

If $\epsilon$ is chosen sufficiently small, depending on $\delta > 0$, then this shows that $u_i = (u_i^1,\ldots , u_i^k)$ is a $\delta$-splitting map on $B(p_i, r_0)$, for sufficiently large i.$\square$