Extension of a theorem of Shi and Tam

TL;DR

This paper generalizes a theorem relating boundary mean curvature integrals of compact manifolds with non-negative scalar curvature, using geometric flows, and establishes rigidity conditions without spin assumptions in low dimensions.

Contribution

It extends Shi and Tam's theorem to more general manifolds and removes the spin condition in low dimensions, using the rac{H}{R}-flow for proof.

Findings

Proved a generalized boundary mean curvature inequality.

Established rigidity conditions characterizing Euclidean domains.

Removed spin assumption in low-dimensional cases.

Abstract

In this note, we prove the following generalization of a theorem of Shi and Tam \cite{ShiTam02}: Let $(\Omega, g)$ be an $n$-dimensional ($n \geq 3$) compact Riemannian manifold, spin when $n>7$, with non-negative scalar curvature and mean convex boundary. If every boundary component $\Sigma_i$ has positive scalar curvature and embeds isometrically as a mean convex star-shaped hypersurface ${\hat \Sigma}_i \subset \R^n$, then \int_{\Sigma_i} H d \sigma \le \int_{{\hat \Sigma}_i} \hat{H} d {\hat \sigma} where $H$ is the mean curvature of $\Sigma_i$ in $(\Omega, g)$, $\hat{H}$ is the Euclidean mean curvature of ${\hat \Sigma}_i$ in $\R^n$, and where $d \sigma$ and $d {\hat \sigma}$ denote the respective volume forms. Moreover, equality in (\ref{eqn: main theorem}) holds for some boundary component $\Sigma_i$ if, and only if, $(\Omega, g)$ is isometric to a domain in $\R^n$. In the proof, we make use of a foliation of the exterior of the $\hat \Sigma_i$'s in $\R^n$ by the $\frac{H}{R}$-flow studied by Gerhardt \cite{Gerhardt90} and Urbas \cite{Urbas90}. We also carefully establish the rigidity statement in low dimensions without the spin assumption that was used in \cite{ShiTam02}