Extension of a theorem of Shi and Tam

In this note, we prove the following generalization of a theorem of Shi and Tam \cite{ShiTam02}: Let $(\mathrm{\Omega },g)$ be an $n$-dimensional ($n\ge 3$) compact Riemannian manifold, spin when $n>7$, with non-negative scalar curvature and mean convex boundary. If every boundary component ${\mathrm{\Sigma }}_i$ has positive scalar curvature and embeds isometrically as a mean convex star-shaped hypersurface ${\widehat{\mathrm{\Sigma }}}_i\subset {\mathbb{R}}^n$, then \int_{\Sigma_i} H d \sigma \le \int_{{\hat \Sigma}_i} \hat{H} d {\hat \sigma} where $H$ is the mean curvature of ${\mathrm{\Sigma }}_i$ in $(\mathrm{\Omega },g)$, $\widehat{H}$ is the Euclidean mean curvature of ${\widehat{\mathrm{\Sigma }}}_i$ in ${\mathbb{R}}^n$, 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 ${\mathrm{\Sigma }}_i$ if, and only if, $(\mathrm{\Omega },g)$ is isometric to a domain in ${\mathbb{R}}^n$. In the proof, we make use of a foliation of the exterior of the $\hat \Sigma_i${}^{\mathrm{\prime }}sin$\R^n$bythe$\frac{H}{R}$-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}