Rigidity for critical metrics of the volume functional

Barros, A.; Silva, A. Da

doi:10.48550/arxiv.1706.07367

Cited by 3 publications

(2 citation statements)

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“…Moreover, the equality holds if and only if M 3 is isometric to a geodesic ball in R 3 or S 3 . This result also holds for negative scalar curvature, provided that the mean curvature of the boundary satisfies H > 2, as was proven in [4]; see also [6]. Another upper bound estimate for the area of the boundary was obtained by Corvino, Eichmair and Miao (cf.…”

Section: Introductionmentioning

confidence: 54%

Isoperimetric inequality and Weitzenböck type formula for critical metrics of the volume

2019

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We provide an isoperimetric inequality for critical metrics of the volume functional with nonnegative scalar curvature on compact manifolds with boundary. In addition, we establish a Weitzenböck type formula for critical metrics of the volume functional on four-dimensional manifolds. As an application, we obtain a classification result for such metrics. Date: December 1, 2017. 2010 Mathematics Subject Classification. Primary 53C25, 53C20, 53C21; Secondary 53C65. Key words and phrases. Volume functional; critical metrics; isoperimetric inequality; Weitzenböck formula. H. Baltazar was partially supported by CNPq/Brazil and FAPEPI/Brazil. E. Ribeiro Jr was partially supported by grants from CNPq/Brazil (Grant: 303091/2015-0), PRONEX-FUNCAP/CNPq/Brazil and CAPES/ Brazil -Finance Code 001.

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Section: Introductionmentioning

confidence: 54%

Isoperimetric inequality and Weitzenböck type formula for critical metrics of the volume

2019

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“…Such result was improved by the first author, Batista, and Bezerra for a generic manifold in an arbitrary dimension (see [4,Theorem 2]). Meanwhile, Barros, and Da Silva presented in [6], an upper bound for the area of the boundary of a compact n-dimensional oriented Miao-Tam critical metric (see also [5,9]). For more references on the critical metrics of the volume functional, see [2,3,6,7,9,20,25], and references therein.…”

Section: Introductionmentioning

confidence: 99%

Weakly Einstein critical metrics of the volume functional on compact manifolds with boundary

Baltazar

Silva

Oliveira

2020

Journal of Mathematical Analysis and Applications

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The goal of this paper is to study weakly Einstein critical metrics of the volume functional on a compact manifold M with smooth boundary ∂M . Here, we will give the complete classification for an n-dimensional, n = 3 or 4, weakly Einstein critical metric of the volume functional with nonnegative scalar curvature. Moreover, in the higher dimensional case (n ≥ 5), we will established a similar result for weakly Einstein critical metric under a suitable constraint on the Weyl tensor.2010 Mathematics Subject Classification. Primary 53C25; Secondary 53C20, 53C21, 53C24.

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Weakly Einstein critical metrics of the volume functional on compact manifolds with boundary

Baltazar

Silva

Oliveira

2018

Preprint

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Rigidity for critical metrics of the volume functional

Cited by 3 publications

References 14 publications

Isoperimetric inequality and Weitzenböck type formula for critical metrics of the volume

Isoperimetric inequality and Weitzenböck type formula for critical metrics of the volume

Weakly Einstein critical metrics of the volume functional on compact manifolds with boundary

Weakly Einstein critical metrics of the volume functional on compact manifolds with boundary

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