Critical metrics of the volume functional on compact three-manifolds with smooth boundary

TL;DR

This paper investigates Miao-Tam critical metrics on compact three-manifolds with boundary, providing boundary area estimates and a Bochner-type formula, leading to classification results for metrics with positive scalar curvature.

Contribution

It introduces boundary area estimates and a Bochner-type formula for Miao-Tam critical metrics, characterizing those with positive scalar curvature as geodesic balls in S^3.

Findings

Boundary area estimates for Miao-Tam critical metrics

A Bochner-type formula for these metrics

Classification of positive scalar curvature cases as geodesic balls in S^3

Abstract

We study the space of smooth Riemannian structures on compact three-manifolds with boundary that satisfies a critical point equation associated with a boundary value problem, for simplicity, Miao-Tam critical metrics. We provide an estimate to the area of the boundary of Miao-Tam critical metrics on compact three-manifolds. In addition, we obtain a B\"ochner type formula which enables us to show that a Miao-Tam critical metric on a compact three-manifold with positive scalar curvature must be isometric to a geodesic ball in $\Bbb{S}^3.$