Critical Metrics of the Volume Functional on Manifolds with Boundary

TL;DR

This paper investigates critical metrics of the volume functional on manifolds with boundary, showing that those with parallel Ricci tensor are isometric to geodesic balls in space forms, thus characterizing their geometric structure.

Contribution

It provides an integral formula and a classification result for critical metrics with parallel Ricci tensor on manifolds with boundary.

Findings

Critical metrics with parallel Ricci tensor are isometric to geodesic balls in space forms.

The paper establishes an integral formula relating the volume functional to geometric properties.

Characterizes the structure of critical metrics on manifolds with boundary under specific curvature conditions.

Abstract

The goal of this article is to study the space of smooth Riemannian structures on compact manifolds with boundary that satisfies a critical point equation associated with a boundary value problem. We provide an integral formula which enables us to show that if a critical metric of the volume functional on a connected $n$-dimensional manifold $M^n$ with boundary $\partial M$ has parallel Ricci tensor, then $M^n$ is isometric to a geodesic ball in a simply connected space form $\mathbb{R}^{n}$, $\mathbb{H}^{n}$ or $\mathbb{S}^{n}$.