Extension of a theorem of Shi and Tam

Michael Eichmair, Pengzi Miao, Xiaodong Wang

Research output: Contribution to journalArticle

4 Citations (Scopus)

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 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 languageEnglish (US)
Pages (from-to)45-56
Number of pages12
JournalCalculus of Variations and Partial Differential Equations
Volume43
Issue number1-2
DOIs
StatePublished - Jan 2012

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Rigidity
Stars
Mean Curvature
Theorem
Positive Scalar Curvature
Nonnegative Curvature
Scalar Curvature
Foliation
Isometric
Compact Manifold
Hypersurface
Riemannian Manifold
n-dimensional
Euclidean
Star
Equality
If and only if
Denote
Generalization
Form

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

Cite this

Extension of a theorem of Shi and Tam. / Eichmair, Michael; Miao, Pengzi; Wang, Xiaodong.

In: Calculus of Variations and Partial Differential Equations, Vol. 43, No. 1-2, 01.2012, p. 45-56.

Research output: Contribution to journalArticle

Eichmair, Michael ; Miao, Pengzi ; Wang, Xiaodong. / Extension of a theorem of Shi and Tam. In: Calculus of Variations and Partial Differential Equations. 2012 ; Vol. 43, No. 1-2. pp. 45-56.
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