Critical Metrics of the Volume Functional on Compact Three-Manifolds with Smooth Boundary

Batista, R.; Diógenes, R.; Ranieri, M.; Ribeiro, E.

doi:10.1007/s12220-016-9730-y

Cited by 43 publications

(59 citation statements)

References 21 publications

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“…The CPE conjecture is true for 3-dimensional manifolds with nonnegative sectional curvature. Now, motivated by the recent work on critical metrics of the volume functional and positive static triples due to first author and Ribeiro Jr. (see [3], Corollaries 1 and 2), see also the works [1,6] for the three-dimensional case, we shall obtain similar result for CPE metrics satisfying the zero radial Weyl curvature assumption. More precisely, we have: Theorem 2.…”

Section: Conjecture 1 a Cpe Metric Is Always Einsteinmentioning

confidence: 60%

On critical point equation of compact manifolds with zero radial Weyl curvature

Baltazar

2018

Geom Dedicata

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Let C be the space of smooth metrics g on a given compact manifold M n (n ≥ 3) with constant scalar curvature and unitary volume. The goal of this paper is to study the critical point of the total scalar curvature functional restricted to the space C (we shall refer to this critical point as CPE metrics) under assumption that (M, g) has zero radial Weyl curvature.Among the results obtained, we emphasize that in 3-dimension we will be able to prove that a CPE metric with nonnegative sectional curvature must be isometric to a standard 3-sphere. We will also prove that a n-dimensional, 4 ≤ n ≤ 10, CPE metric satisfying a L n/2 -pinching condition will be isometric to a standard sphere. In addition, we shall conclude that such critical metrics are isometrics to a standard sphere under fourth-order vanishing condition on the Weyl tensor.

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Section: Conjecture 1 a Cpe Metric Is Always Einsteinmentioning

confidence: 60%

On critical point equation of compact manifolds with zero radial Weyl curvature

Baltazar

2018

Geom Dedicata

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“…We refer to [22] for a general discussion on this topic. Inspired by a classical result obtained in [10] and [24], it has been shown by Batista, Ranieri and the last two named authors [7] that the boundary ∂M of a compact three-dimensional oriented Miao-Tam critical metric (M 3 , g) with connected boundary and nonnegative scalar curvature must be a 2-sphere whose area satisfies the inequality…”

Section: Introductionmentioning

confidence: 99%

“…Let (M n , g) be a connected compact Riemannian manifold with dimension n at least three and smooth boundary ∂M. According to [5,7,14,20] and [21], we say that g is, for simplicity, a Miao-Tam critical metric if there is a nonnegative smooth function f on M n such that f −1 (0) = ∂M and satisfies the overdetermined-elliptic system…”

Section: Introductionmentioning

confidence: 99%

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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“…where Ric g and Hess g f stand, respectively, for the Ricci tensor and the Hessian of f associated to g on M n (see [23] for more details). Hence, following the terminology used in [2,8,9,24] we recall the definition of Miao-Tam critical metric.…”

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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Critical Metrics of the Volume Functional on Compact Three-Manifolds with Smooth Boundary

Cited by 43 publications

References 21 publications

On critical point equation of compact manifolds with zero radial Weyl curvature

On critical point equation of compact manifolds with zero radial Weyl curvature

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

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