On the Stability of K\"ahler-Einstein Metrics

TL;DR

This paper proves the stability of Kähler-Einstein metrics with nonpositive scalar curvature using spin$^c$ structures and explores their extremal properties related to the Yamabe invariant and volume among metrics with similar scalar curvature.

Contribution

It establishes the stability of certain Kähler-Einstein metrics and links their geometric properties to the Yamabe invariant under specific deformation conditions.

Findings

Kähler-Einstein metrics with nonpositive scalar curvature are stable under conformal changes.

Such metrics are local maxima of the Yamabe invariant if all infinitesimal complex deformations are integrable.

These metrics have minimal volume among metrics with scalar curvature at least as large.

Abstract

Using spin$^c$ structure we prove that K\"ahler-Einstein metrics with nonpositive 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 deformation 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\"ahler-Einstein metric.