### 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 Σ_{i} ⊂ ℝ^{n}, then, where H is the mean curvature of Σ_{i} in (Ω, g), H is the Euclidean mean curvature of Σ_{i} in ℝ^{n}, and where d σ and dσ 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 ℝ^{n}. In the proof, we make use of a foliation of the exterior of the Σ_{i}'s in ℝ^{n} 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) |
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Pages (from-to) | 45-56 |

Number of pages | 12 |

Journal | Calculus of Variations and Partial Differential Equations |

Volume | 43 |

Issue number | 1-2 |

DOIs | |

State | Published - Jan 1 2012 |

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

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## Cite this

*Calculus of Variations and Partial Differential Equations*,

*43*(1-2), 45-56. https://doi.org/10.1007/s00526-011-0402-2