## Journal of Differential Geometry

### 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).

#### Article information

Source
J. Differential Geom., Volume 93, Number 2 (2013), 175-189.

Dates
First available in Project Euclid: 25 February 2013

https://projecteuclid.org/euclid.jdg/1361800864

Digital Object Identifier
doi:10.4310/jdg/1361800864

Mathematical Reviews number (MathSciNet)
MR3024304

Zentralblatt MATH identifier
1279.32020

#### Citation

Chen, X.-X.; Donaldson, S. K. Volume estimates for Kähler-Einstein metrics: The three-dimensional case. J. Differential Geom. 93 (2013), no. 2, 175--189. doi:10.4310/jdg/1361800864. https://projecteuclid.org/euclid.jdg/1361800864