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On the Stability of Kahler-Einstein Metrics Xianzhe Dai

Summary: On the Stability of Kšahler-Einstein Metrics
Xianzhe Dai
Xiaodong Wang
Guofang Wei
April 19, 2005
Using spinc
structure we prove that Kšahler-Einstein 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 Kšahler-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 Kšahler-Einstein 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
non-positive in the direction of changes in conformal structures (we have weakened the notion by
allowing kernels). By the well-known formula, this is to say,


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


Collections: Mathematics