### Abstract

We study the volume functional on the space of constant scalar curvature metrics with a prescribed boundary metric. We derive a sufficient and necessary condition for a metric to be a critical point, and show that the only domains in space forms, on which the standard metrics are critical points, are geodesic balls. In the zero scalar curvature case, assuming the boundary can be isometrically embedded in the Euclidean space as a compact strictly convex hypersurface, we show that the volume of a critical point is always no less than the Euclidean volume bounded by the isometric embedding of the boundary, and the two volumes are equal if and only if the critical point is isometric to a standard Euclidean ball. We also derive a second variation formula and apply it to show that, on Euclidean balls and "small" hyperbolic and spherical balls in dimensions 3 ≤ n ≤ 5, the standard space form metrics are indeed saddle points for the volume functional.

Original language | English (US) |
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Pages (from-to) | 141-171 |

Number of pages | 31 |

Journal | Calculus of Variations and Partial Differential Equations |

Volume | 36 |

Issue number | 2 |

DOIs | |

State | Published - Sep 2009 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

**On the volume functional of compact manifolds with boundary with constant scalar curvature.** / Miao, Pengzi; Tam, Luen F.

Research output: Contribution to journal › Article

*Calculus of Variations and Partial Differential Equations*, vol. 36, no. 2, pp. 141-171. https://doi.org/10.1007/s00526-008-0221-2

}

TY - JOUR

T1 - On the volume functional of compact manifolds with boundary with constant scalar curvature

AU - Miao, Pengzi

AU - Tam, Luen F.

PY - 2009/9

Y1 - 2009/9

N2 - We study the volume functional on the space of constant scalar curvature metrics with a prescribed boundary metric. We derive a sufficient and necessary condition for a metric to be a critical point, and show that the only domains in space forms, on which the standard metrics are critical points, are geodesic balls. In the zero scalar curvature case, assuming the boundary can be isometrically embedded in the Euclidean space as a compact strictly convex hypersurface, we show that the volume of a critical point is always no less than the Euclidean volume bounded by the isometric embedding of the boundary, and the two volumes are equal if and only if the critical point is isometric to a standard Euclidean ball. We also derive a second variation formula and apply it to show that, on Euclidean balls and "small" hyperbolic and spherical balls in dimensions 3 ≤ n ≤ 5, the standard space form metrics are indeed saddle points for the volume functional.

AB - We study the volume functional on the space of constant scalar curvature metrics with a prescribed boundary metric. We derive a sufficient and necessary condition for a metric to be a critical point, and show that the only domains in space forms, on which the standard metrics are critical points, are geodesic balls. In the zero scalar curvature case, assuming the boundary can be isometrically embedded in the Euclidean space as a compact strictly convex hypersurface, we show that the volume of a critical point is always no less than the Euclidean volume bounded by the isometric embedding of the boundary, and the two volumes are equal if and only if the critical point is isometric to a standard Euclidean ball. We also derive a second variation formula and apply it to show that, on Euclidean balls and "small" hyperbolic and spherical balls in dimensions 3 ≤ n ≤ 5, the standard space form metrics are indeed saddle points for the volume functional.

UR - http://www.scopus.com/inward/record.url?scp=70350165192&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=70350165192&partnerID=8YFLogxK

U2 - 10.1007/s00526-008-0221-2

DO - 10.1007/s00526-008-0221-2

M3 - Article

AN - SCOPUS:70350165192

VL - 36

SP - 141

EP - 171

JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

SN - 0944-2669

IS - 2

ER -