Stability of the L^p-Norm of the Curvature Tensor at Kahler Space Forms

TL;DR

This paper investigates the stability of the L^p-norm of the curvature tensor on Kahler manifolds, showing that constant holomorphic sectional curvature metrics locally minimize this functional.

Contribution

It establishes that on Kahler manifolds, the L^p-norm of the curvature tensor attains local minima at metrics with constant holomorphic sectional curvature.

Findings

L^p-norm functional has local minima at constant holomorphic sectional curvature metrics.

The result applies to the space of Kahler metrics with fixed volume.

Provides insight into the geometric stability of Kahler space forms.

Abstract

We consider the Riemannian functional defined on the space of Riemannian metrics with unit volume on a closed smooth manifold M given by $R_p(g) :=\int_M|R(g)|^pdvg$ where $R(g)$, $dv_g$ denote the corresponding Riemannian curvature, volume form and p is a real number greater than or equal to 2. We prove that $R_p$ restricted to the space of Kahler metrics attains its local minima at a metric with constant holomorphic sectional curvature.