Critical metrics of the volume functional with pinched curvature

TL;DR

This paper characterizes critical metrics of the volume functional with pinched Weyl curvature, showing they are isometric to geodesic balls in spheres and providing conditions for classification based on the potential function.

Contribution

It establishes a classification of critical volume metrics with pinched Weyl curvature and introduces a condition involving the potential function's gradient.

Findings

Critical metrics are isometric to geodesic balls in spheres.

A necessary and sufficient condition on the potential function's gradient is provided.

The results extend understanding of volume functional critical points with curvature constraints.

Abstract

In this paper, we prove that a critical metric of the volume functional with pinched Weyl curvature is isometric to a geodesic ball in $\mathbb{S}^{n}.$ Moreover, we provide a necessary and sufficient condition on the norm of the gradient of the potential function in order to classify such critical metrics.