## Volume estimates for Kähler-Einstein metrics: the three-dimensional case.(English)Zbl 1279.32020

The authors provide some volume estimates of neighborhoods of sets of big curvature for a Kähler-Einstein threefold.
Let $$(M,g)$$ be a compact Kähler manifold of dimension 3 and fix $$r>0$$. Define $$K_r$$ to bethe subset of $$M$$ consisting of points where $$| \text{Riem} | \geq 1/r^{2}$$. Let $$Z_r$$ be the $$r$$-neighborhood of $$K_r$$. Let us assume now that $$g$$ is a Kähler-Einstein metric with Einstein constant $$=1$$, so $$M$$ is a Fano Kähler-Einstein manifold. Then, in a nutshell, the authors prove a volume estimate of the form
$\text{Vol}(Z_r)\leq Cr^4,$ where $$C$$ depends only on some topological data. Note that it is not too hard to check that the volume of $$K_r$$ enjoys a similar inequality.
The previous theorem is based on the study of the associated $$L^2$$ energy of the underlying region. More precisely, consider the normalized energy over a ball centered at $$x\in M$$,
$E(x,r)=r^{-2} \int_{B(x,r)} | \text{Riem}|^2 dV$ where $$x\in M$$. Then, in the same context as above, the authors show that $$E(x,r)$$ enjoys an approximate monotonicity property for the variable $$r$$ in the following sense. For each $$\epsilon >0$$, there exists $$\delta>0$$ such that for each metric ball $$B(x,r)$$ which does not carry homology and with $$E(x,r)\leq \delta$$, for any $$y\in B(x,r/2)$$ and for each $$r'\leq r/2$$, one has $$E(y,r')\leq \epsilon$$. Note that the authors provide some hints about the function $$\delta(\epsilon)$$. In other words, the normalized energy of a ball $$B$$ controls the normalized energy in an interior ball $$\subset B$$, when one assumes that the inclusion map $$H_2(\partial B,\mathbb{R}) \rightarrow H_2(B,\mathbb{R})$$ is surjective (actually this latter technical assumption can be removed).
In a different paper [J. Differ. Geom. 93, No. 2, 191–201 (2013; Zbl 1281.32019)], the authors strengthen and extend the results of the paper under review to Kähler-Einstein manifolds of any dimension.
As explained in the visionary paper [S. K. Donaldson, Discussion of the Kähler-Einstein problem”, http://www2.imperial.ac.uk/~skdona/KENOTES.PDF], these volume estimates are crucial to understand the structure of Gromov-Hausdorff limit spaces of Einstein Fano manifolds.

### MSC:

 32Q20 Kähler-Einstein manifolds 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions

Zbl 1281.32019
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