### 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) |
---|---|

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 2012 |

### Fingerprint

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

*Calculus of Variations and Partial Differential Equations*,

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

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

Research output: Contribution to journal › Article

*Calculus of Variations and Partial Differential Equations*, vol. 43, no. 1-2, pp. 45-56. https://doi.org/10.1007/s00526-011-0402-2

}

TY - JOUR

T1 - Extension of a theorem of Shi and Tam

AU - Eichmair, Michael

AU - Miao, Pengzi

AU - Wang, Xiaodong

PY - 2012/1

Y1 - 2012/1

N2 - 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).

AB - 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).

UR - http://www.scopus.com/inward/record.url?scp=81755163908&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=81755163908&partnerID=8YFLogxK

U2 - 10.1007/s00526-011-0402-2

DO - 10.1007/s00526-011-0402-2

M3 - Article

VL - 43

SP - 45

EP - 56

JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

SN - 0944-2669

IS - 1-2

ER -