Critical metrics of the scalar curvature functional satisfying a vanishing condition on the Weyl tensor

Santos, Alex Sandro

doi:10.1007/s00013-017-1030-7

Cited by 5 publications

(4 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Then we will able to prove a rigidity result for such metrics in dimension n, n ≥ 4, provided that our manifold satisfies div 4 W = 0. This result was motivated by the work of Qing and Yuan [35] as well as by the very recent article of Santos [36]. In order to do this, it is worth reporting the following formula for the divergence of the Bach tensor (see [9] for more details).…”

Section: Proof In What Follows We Takementioning

confidence: 99%

“…In despite some important progresses, it remains a big challenge to prove the Besse's conjecture. For more references on CPE metrics, see [7,14,28,30,35,36] and references therein.…”

Section: Conjecture 1 a Cpe Metric Is Always Einsteinmentioning

confidence: 99%

“…Recently, based in the ideas developed in [13], Santos obtained a positive answer for Conjecture 1 under the second order divergence-free Weyl tensor condition (cf. [36,Theorem 1] for more details). Thus, inspired in the previous results, it is natural to ask what happens in higher dimension for a CPE metric satisfying the fourth order divergence-free Weyl tensor condition.…”

Section: Conjecture 1 a Cpe Metric Is Always Einsteinmentioning

confidence: 99%

See 2 more Smart Citations

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

Baltazar

2018

Geom Dedicata

View full text Add to dashboard Cite

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.

show abstract

Section: Proof In What Follows We Takementioning

confidence: 99%

“…In despite some important progresses, it remains a big challenge to prove the Besse's conjecture. For more references on CPE metrics, see [7,14,28,30,35,36] and references therein.…”

Section: Conjecture 1 a Cpe Metric Is Always Einsteinmentioning

confidence: 99%

Section: Conjecture 1 a Cpe Metric Is Always Einsteinmentioning

confidence: 99%

See 1 more Smart Citation

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

Baltazar

2018

Geom Dedicata

View full text Add to dashboard Cite

show abstract

“…Notice that the fourth-order divergence-free Weyl tensor assumption is clearly weaker than locally conformally flat and harmonic Weyl tensor conditions considered in [20,26]. Moreover, this assumption was recently used to classify critical metrics of the volume functional, static spaces and CPE metrics, for more details see, e.g., [3,4,19,35,38]. It is also important to emphasize that Example 1 has fourth-order divergence-free Weyl tensor, but it has no zero radial Weyl curvature.…”

Section: Introductionmentioning

confidence: 99%

Remarks on compact quasi-Einstein manifolds with boundary

Diógenes¹,

Gadelha²,

Ribeiro³

2021

Preprint

View full text Add to dashboard Cite

In this paper, we prove that a compact quasi-Einstein manifold (M n , g, u) of dimension n ≥ 4 with boundary ∂M, nonnegative sectional curvature and zero radial Weyl tensor is either isometric, up to scaling, to the standard hemisphere S n + , or g = dt 2 + ψ 2 (t)g L and u = u(t), where g L is Einstein with nonnegative Ricci curvature. A similar classification result is obtained by assuming a fourth-order vanishing condition on the Weyl tensor. Moreover, a new example is presented in order to justify our assumptions. In addition, the case of dimension n = 3 is also discussed.

show abstract

Besse conjecture for compact manifolds with pinched curvature

Baltazar

2020

Arch. Math.

View full text Add to dashboard Cite

Critical metrics of the scalar curvature functional satisfying a vanishing condition on the Weyl tensor

Cited by 5 publications

References 12 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

Remarks on compact quasi-Einstein manifolds with boundary

Besse conjecture for compact manifolds with pinched curvature

Contact Info

Product

Resources

About