Remarks on critical metrics of the scalar curvature and volume functionals on compact manifolds with boundary

Abstract

We provide a general Böchner type formula which enables us to prove some rigidity results for $V$-static spaces. In particular, we show that an $n$-dimensional positive static triple with connected boundary and positive scalar curvature must be isometric to the standard hemisphere, provided that the metric has zero radial Weyl curvature and satisfies a suitable pinching condition. Moreover, we classify $V$-static spaces with non-negative sectional curvature.

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References in corpus (3)

• On the volume functional of compact manifolds with boundary with constant scalar curvature
• Uniqueness of the de Sitter spacetime among static vacua with positive cosmological constant
• On static three-manifolds with positive scalar curvature

Cited by in corpus (10)

• On the mass of static metrics with positive cosmological constant - I
• Quantum reverse hypercontractivity: its tensorization and application to strong converses
• On the mass of static metrics with positive cosmological constant -- II
• Rigidity for critical metrics of the volume functional
• Weakly Einstein critical metrics of the volume functional on compact manifolds with boundary
• Critical metrics of the volume functional on three-dimensional manifolds
• On Static Manifolds and Related Critical Spaces with cyclic parallel Ricci tensor
• Isoperimetric inequality and Weitzenböck type formula for critical metrics of the volume
• Geometric inequalities for critical metrics of the volume functional
• Volume functional of compact $4$-manifolds with a prescribed boundary metric