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- W2170095180 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 Σ i has positive scalar curvature and embeds isometrically as a mean convex star-shaped hypersurface $${{hat Sigma}_i subset mathbb{R}^n}$$ , then $$ intlimits_{Sigma_i} H d sigma le intlimits_{{hat Sigma}_i} hat{H} d {hat sigma} $$ where H is the mean curvature of Σ i 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 Σ i 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)." @default.
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- W2170095180 date "2011-02-16" @default.
- W2170095180 modified "2023-09-26" @default.
- W2170095180 title "Extension of a theorem of Shi and Tam" @default.
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- W2170095180 doi "https://doi.org/10.1007/s00526-011-0402-2" @default.
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