Critical metrics of the volume functional on manifolds with boundary

Baltazar, H.; Ribeiro, E.

doi:10.1090/proc/13619

Cited by 28 publications

(26 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Recently, Baltazar and Ribeiro were able to improve Theorem by showing that the Einstein assumption on

(M^{n}, g)

can be replaced by the parallel Ricci tensor condition, which is weaker than the former. Another recent result obtained about Miao–Tam critical metrics are volume estimates of the boundary of such manifolds.…”

Section: Introductionmentioning

confidence: 99%

Rigidity for critical metrics of the volume functional

Barros

Silva

2018

Mathematische Nachrichten

View full text Add to dashboard Cite

Geodesic balls in a simply connected space forms Sn, Rn or Hn are distinguished manifolds for comparison in bounded Riemannian geometry. In this paper we show that they have the maximum possible boundary volume among Miao–Tam critical metrics with connected boundary provided that the boundary of the manifold has a lower bound for the Ricci curvature. In the same spirit we also extend a rigidity theorem due to Boucher et al. and Shen to n‐dimensional static metrics with positive constant scalar curvature, which gives us a partial answer to the Cosmic no‐hair conjecture.

show abstract

“…Recently, Baltazar and Ribeiro were able to improve Theorem by showing that the Einstein assumption on

(M^{n}, g)

Section: Introductionmentioning

confidence: 99%

Rigidity for critical metrics of the volume functional

Barros

Silva

2018

Mathematische Nachrichten

View full text Add to dashboard Cite

show abstract

“…In fact, they proved that any connected, compact, Einstein manifold with smooth boundary satisfying Miao-Tam critical condition is isometric to a geodesic ball in a simply connected space form. And then several generalizations of this rigidity result were found by different authors, replacing the Einstein assumption by a weaker condition such as harmonic Weyl tensor [3], parallel Ricci tensor [4], or cyclic parallel Ricci tensor [5]. For Some other generalizations or rigidity results, we can refer to [6][7][8][9][10], etc.…”

Section: Introductionmentioning

confidence: 85%

Paracontact Metric $(κ, μ)$ -Manifold Satisfying the Miao-Tam Equation

Yin

2021

Advances in Mathematical Physics

View full text Add to dashboard Cite

In this paper, we classified the paracontact metric κ , μ -manifold satisfying the Miao-Tam critical equation with κ > − 1 . We proved that it is locally isometric to the product of a flat n + 1 -dimensional manifold and an n -dimensional manifold of negative constant curvature − 4 .

show abstract

“…Later, in [33], they classified all Einstein or conformally flat metrics which are critical points for the volume functional restrict to the above space. See also the works [2,4,19,32,33] and references therein.…”

Section: Introductionmentioning

confidence: 99%

On the $σ_2$-curvature and volume of compact manifolds

Andrade¹,

Tiarlos²,

Santos³

2021

Preprint

View full text Add to dashboard Cite

In this work we are interested in studying deformations of the σ2-curvature and the volume. For closed manifolds, we relate critical points of the total σ2-curvature functional to the σ2-Einstein metrics and, as a consequence of results of H. J. Gursky and J. A. Viaclovsky [25] and Z. Hu and H. Li [27], we obtain a sufficient and necessary condition for a critical metric to be Einstein. Moreover, we show a volume comparison result for Einstein manifolds with respect to σ2-curvature which shows that the volume can be controlled by the σ2-curvature under certain conditions. Next, for compact manifold with nonempty boundary, we study variational properties of the volume functional restricted to the space of metrics with constant σ2-curvature and with fixed induced metric on the boundary. We characterize the critical points to this functional as the solutions of an equation and show that in space forms they are geodesic balls. Studying second order properties of the volume functional we show that there is a variation for which geodesic balls are indeed local minimum in a natural direction.

show abstract

Critical metrics of the volume functional on manifolds with boundary

Cited by 28 publications

References 18 publications

Rigidity for critical metrics of the volume functional

Rigidity for critical metrics of the volume functional

Paracontact Metric $(κ, μ)$ -Manifold Satisfying the Miao-Tam Equation

On the $σ_2$-curvature and volume of compact manifolds

Contact Info

Product

Resources

About

Critical metrics of the volume functional on manifolds with boundary

Cited by 28 publications

References 18 publications

Rigidity for critical metrics of the volume functional

Rigidity for critical metrics of the volume functional

Paracontact Metric κ , μ -Manifold Satisfying the Miao-Tam Equation

On the $σ_2$-curvature and volume of compact manifolds

Contact Info

Product

Resources

About

Paracontact Metric $(κ, μ)$ -Manifold Satisfying the Miao-Tam Equation