Critical metrics of the volume functional on manifolds with boundary

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 $\mathbb {S}^{n}$.