Extension of a theorem of Shi and Tam

Abstract

In this note, we prove the following generalization of a theorem of Shi and Tam (J Differ Geom 62:79–125, 2002): Let (Ω, g) be an n-dimensional (n ≥ 3) compact Riemannian manifold, spin when n > 7, with non-negative scalar curvature and mean convex boundary. If every boundary component Σi has positive scalar curvature and embeds isometrically as a mean convex star-shaped hypersurface \({{\hat \Sigma}_i \subset \mathbb{R}^n}\), then $$ \int\limits_{\Sigma_i} H d \sigma \le \int\limits_{{\hat \Sigma}_i} \hat{H} d {\hat \sigma} $$ where H is the mean curvature of Σi in (Ω, g), \({\hat{H}}\) is the Euclidean mean curvature of \({{\hat \Sigma}_i}\) in \({\mathbb{R}^n}\), and where d σ and \({d {\hat \sigma}}\) denote the respective volume forms. Moreover, equality holds for some boundary component Σi if, and only if, (Ω, g) is isometric to a domain in \({\mathbb{R}^n}\). In the proof, we make use of a foliation of the exterior of the \({\hat \Sigma_i}\)’s in \({\mathbb{R}^n}\) by the \({\frac{H}{R}}\)-flow studied by Gerhardt (J Differ Geom 32:299–314, 1990) and Urbas (Math Z 205(3):355–372, 1990). We also carefully establish the rigidity statement in low dimensions without the spin assumption that was used in Shi and Tam (J Differ Geom 62:79–125, 2002).