TITLE

Extension of a theorem of Shi and Tam

AUTHOR(S)
Eichmair, Michael; Miao, Pengzi; Wang, Xiaodong
PUB. DATE
January 2012
SOURCE
Calculus of Variations & Partial Differential Equations;Jan2012, Vol. 43 Issue 1/2, p45
SOURCE TYPE
Academic Journal
DOC. TYPE
Article
ABSTRACT
In this note, we prove the following generalization of a theorem of Shi and Tam (J Differ Geom 62:79-125, ): Let (O, 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 S has positive scalar curvature and embeds isometrically as a mean convex star-shaped hypersurface $${{\hat \Sigma}_i \subset \mathbb{R}^n}$$, thenwhere H is the mean curvature of S in (O, g), $${\hat{H}}$$ is the Euclidean mean curvature of $${{\hat \Sigma}_i}$$ in $${\mathbb{R}^n}$$, and where d s and $${d {\hat \sigma}}$$ denote the respective volume forms. Moreover, equality holds for some boundary component S if, and only if, (O, 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, ) and Urbas (Math Z 205(3):355-372, ). 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, ).
ACCESSION #
67363493

 

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