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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

Citations:

Zbl 1281.32019
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