On the variational stability of Kähler-Einstein metrics

Using $spin^c$ structure we prove that Kähler-Einstein 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 deformations of the complex structure are integrable, then the Kähler- 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 Kähler-Einstein metric.

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Published 1 January 2007