Volume estimates for Kähler-Einstein metrics: The three-dimensional case

Abstract

We obtain an estimate for the volumes of neighborhoods of sets of large curvature in three-dimensional Kähler-Einstein manifolds. The key technical step is to prove that a version of monotonicity for $L^2$ energy holds as long as the underlying region does not “carry homology” (in the sense that the normalized energy in a ball controls the normalized energy in an interior ball).