On Ricci curvature and volume growth in dimension three

Reiris, Martin

doi:10.4310/jdg/1421415565

Cited by 5 publications

(2 citation statements)

References 27 publications

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“…Understanding metric and topological consequences of curvature decay conditions is a very interesting and widely studied problem in geometric analysis. Dropping any attempt of being complete, we refer the interested reader to the aforementioned [6], to the seminal [16], to [65], where the case n = 3 is considered, and to [76] and the references therein.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature

Agostiniani¹,

Fogagnolo²,

Mazzieri³

2018

Preprint

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In this paper we consider complete noncompact Riemannian manifolds (M, g) with nonnegative Ricci curvature and Euclidean volume growth, of dimension n ≥ 3. For every bounded open subset Ω ⊂ M with smooth boundary, we prove thatwhere H is the mean curvature of ∂Ω and AVR(g) is the asymptotic volume ratio of (M, g). Moreover, the equality holds true if and only if (M \Ω, g) is isometric to a truncated cone over ∂Ω. An optimal version of Huisken's Isoperimetric Inequality for 3-manifolds is obtained using this result. Finally, exploiting a natural extension of our techniques to the case of parabolic manifolds, we also deduce an enhanced version of Kasue's non existence result for closed minimal hypersurfaces in manifolds with nonnegative Ricci curvature.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature

Agostiniani¹,

Fogagnolo²,

Mazzieri³

2018

Preprint

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show abstract

“…flatness were investigated again in [8,9] (see also [10]). We explain these results in some detail as they will be relevant for the rest of the paper.…”

Section: Introductionmentioning

confidence: 99%

The asymptotic of static isolated systems and a generalized uniqueness for Schwarzschild

Reiris

2015

Class. Quantum Grav.

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It is proved that any static system that is spacetime-geodesically complete at infinity, and whose spacelike-topology outside a compact set is that of R 3 minus a ball, is asymptotically flat. The matter is assumed compactly supported and no energy condition is required. A similar (though stronger) result applies to black holes too. This allows us to state a large generalisation of the uniqueness of the Schwarzschild solution not requiring asymptotic flatness. The Korotkin-Nicolai static black-hole shows that, for the given generalisation, no further flexibility in the hypothesis is possible.

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Sharp isoperimetric and Sobolev inequalities in spaces with nonnegative Ricci curvature

Balogh

Kristály

2022

Math. Ann.

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On Ricci curvature and volume growth in dimension three

Abstract: We prove that any complete metric on R 3 minus a ball with nonnegative Ricci curvature and quadratic Ricci-curvature decay, has cubic volume growth.

Cited by 5 publications

References 27 publications

Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature

Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature

The asymptotic of static isolated systems and a generalized uniqueness for Schwarzschild

Sharp isoperimetric and Sobolev inequalities in spaces with nonnegative Ricci curvature

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