Summary: communications in
analysis and geometry
Volume 15, Number 4, 669­693, 2007
On the variational stability of
K¨ahler­Einstein metrics
Xianzhe Dai, Xiaodong Wang and Guofang Wei
Using spinc
structure we prove that K¨ahler­Einstein metrics with
non-positive scalar curvature are stable (in the direction of changes
in conformal structures) as the critical points of the total scalar
curvature functional. Moreover, if all infinitesimal complex defor-
mations of the complex structure are integrable, then the K¨ahler­
Einstein metric is a local maximal of the Yamabe invariant, and its
volume is a local minimum among all metrics with scalar curvature
bigger or equal to the scalar curvature of the K¨ahler­Einstein
metric.
1. Introduction
Stability issue comes up naturally in variational problems. One of the most
important geometric variational problems is that of the total scalar curvature
functional. Following [1, p. 132], we call an Einstein metric stable if the

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

Collections: Mathematics