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 Σi ⊂ ℝn, then, where H is the mean curvature of Σi in (Ω, g), H is the Euclidean mean curvature of Σi in ℝn, and where d σ and dσ 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 ℝn. In the proof, we make use of a foliation of the exterior of the Σi's in ℝn by the 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).
|Original language||English (US)|
|Number of pages||12|
|Journal||Calculus of Variations and Partial Differential Equations|
|State||Published - Jan 1 2012|
ASJC Scopus subject areas
- Applied Mathematics