### Abstract

The stationary points of the total scalar curvature functional on the space of unit volume metrics on a given closed manifold are known to be precisely the Einstein metrics. One may consider the modified problem of finding stationary points for the volume functional on the space of metrics whose scalar curvature is equal to a given constant. In this paper, we localize a condition satisfied by such stationary points to smooth bounded domains. The condition involves a generalization of the static equations, and we interpret solutions (and their boundary values) of this equation variationally. On domains carrying a metric that does not satisfy the condition, we establish a local deformation theorem that allows one to achieve simultaneously small prescribed changes of the scalar curvature and of the volume by a compactly supported variation of the metric. We apply this result to obtain a localized gluing theorem for constant scalar curvature metrics in which the total volume is preserved. Finally, we note that starting from a counterexample of Min-Oo's conjecture such as that of Brendle-Marques-Neves, counterexamples of arbitrarily large volume and different topological types can be constructed.

Original language | English (US) |
---|---|

Pages (from-to) | 551-584 |

Number of pages | 34 |

Journal | Mathematische Annalen |

Volume | 357 |

Issue number | 2 |

DOIs | |

State | Published - Oct 2013 |

### Fingerprint

### Keywords

- Primary 53C21

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Mathematische Annalen*,

*357*(2), 551-584. https://doi.org/10.1007/s00208-013-0903-8

**Deformation of scalar curvature and volume.** / Corvino, Justin; Eichmair, Michael; Miao, Pengzi.

Research output: Contribution to journal › Article

*Mathematische Annalen*, vol. 357, no. 2, pp. 551-584. https://doi.org/10.1007/s00208-013-0903-8

}

TY - JOUR

T1 - Deformation of scalar curvature and volume

AU - Corvino, Justin

AU - Eichmair, Michael

AU - Miao, Pengzi

PY - 2013/10

Y1 - 2013/10

N2 - The stationary points of the total scalar curvature functional on the space of unit volume metrics on a given closed manifold are known to be precisely the Einstein metrics. One may consider the modified problem of finding stationary points for the volume functional on the space of metrics whose scalar curvature is equal to a given constant. In this paper, we localize a condition satisfied by such stationary points to smooth bounded domains. The condition involves a generalization of the static equations, and we interpret solutions (and their boundary values) of this equation variationally. On domains carrying a metric that does not satisfy the condition, we establish a local deformation theorem that allows one to achieve simultaneously small prescribed changes of the scalar curvature and of the volume by a compactly supported variation of the metric. We apply this result to obtain a localized gluing theorem for constant scalar curvature metrics in which the total volume is preserved. Finally, we note that starting from a counterexample of Min-Oo's conjecture such as that of Brendle-Marques-Neves, counterexamples of arbitrarily large volume and different topological types can be constructed.

AB - The stationary points of the total scalar curvature functional on the space of unit volume metrics on a given closed manifold are known to be precisely the Einstein metrics. One may consider the modified problem of finding stationary points for the volume functional on the space of metrics whose scalar curvature is equal to a given constant. In this paper, we localize a condition satisfied by such stationary points to smooth bounded domains. The condition involves a generalization of the static equations, and we interpret solutions (and their boundary values) of this equation variationally. On domains carrying a metric that does not satisfy the condition, we establish a local deformation theorem that allows one to achieve simultaneously small prescribed changes of the scalar curvature and of the volume by a compactly supported variation of the metric. We apply this result to obtain a localized gluing theorem for constant scalar curvature metrics in which the total volume is preserved. Finally, we note that starting from a counterexample of Min-Oo's conjecture such as that of Brendle-Marques-Neves, counterexamples of arbitrarily large volume and different topological types can be constructed.

KW - Primary 53C21

UR - http://www.scopus.com/inward/record.url?scp=84883776881&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84883776881&partnerID=8YFLogxK

U2 - 10.1007/s00208-013-0903-8

DO - 10.1007/s00208-013-0903-8

M3 - Article

VL - 357

SP - 551

EP - 584

JO - Mathematische Annalen

JF - Mathematische Annalen

SN - 0025-5831

IS - 2

ER -