 
Summary: On the Stability of KšahlerEinstein Metrics
Xianzhe Dai
Xiaodong Wang
Guofang Wei
April 19, 2005
Abstract
Using spinc
structure we prove that KšahlerEinstein metrics with nonpositive scalar curva
ture 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 deformation of the com
plex structure are integrable, then the KšahlerEinstein 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 KšahlerEinstein 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 [Bes87, Page 132]
we call an Einstein metric stable if the second variation of the total scalar curvature functional is
nonpositive in the direction of changes in conformal structures (we have weakened the notion by
allowing kernels). By the wellknown formula, this is to say,
