Extension of a theorem of Shi and Tam

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}