A Positive Mass Theorem on Asymptotically Hyperbolic Manifolds with Corners along a Hypersurface

In the asymptotically locally hyperbolic setting it is possible to have metrics with scalar curvature ≥ −6 and negative mass when the genus of the conformal boundary at infinity is positive. Using inverse mean curvature flow, we prove a Penrose inequality for these negative mass metrics. The motivation comes from a previous result of P. Chruściel and W. Simon, which states that the Penrose inequality we prove implies a static uniqueness theorem for negative mass Kottler metrics.

show abstract

“…Although we are unable to replacem by m in general, we do obtain a slightly stronger inequality than(1). See Lemma 3.13.…”

mentioning

confidence: 85%

The Penrose Inequality for Asymptotically Locally Hyperbolic Spaces with Nonpositive Mass

Lee

Neves

2015

Commun. Math. Phys.

View full text Add to dashboard Cite

show abstract

“…Then by the proof of Theorem 1.1 in [3], we may smooth the metric along Σ 2 to get a new AH metric g with the scalar curvature R g −n(n − 1) and mass aspects is negative which is a contradiction to Theorem 1.3 of [1]. This completes the proof of Theorem 3.4.…”

Section: Proof the Main Theoremsmentioning

confidence: 74%

Nonexistence of NNSC-cobordism of Bartnik data

Shi

2020

Preprint

View full text Add to dashboard Cite

In this paper, we consider the problem of nonnegative scalar curvature (NNSC) cobordism of Bartnik data (Σ n−1 1 , γ 1 , H 1 ) and (Σ n−1 2 , γ 2 , H 2 ). We prove that given two metrics γ 1 and γ 2 on S n−1 (3 n 7) with H 1 fixed, then (S n−1 , γ 1 , H 1 ) and (S n−1 , γ 2 , H 2 ) admit no NNSC cobordism provided the prescribed mean curvature H 2 is large enough(Theorem 1.3). Moreover, we show that for n = 3, a much weaker condition that the total mean curvature S 2 H 2 dµγ 2 is large enough rules out NNSC cobordisms(Theorem 1.2); if we require the Gaussian curvature of γ 2 to be positive, we get a criterion for non existence of trivial NNSC-cobordism by using Hawking mass and Brown-York mass(Theorem 1.1). For the general topology case, we prove that (Σ n−1 1 , γ 1 , 0) and (Σ n−1 2 , γ 2 , H 2 ) admit no NNSC cobordism provided the prescribed mean curvature H 2 is large enough(Theorem 1.5).

show abstract

“…This is essentially a consequence of the positive mass theorem on asymptotically hyperbolic manifolds (see [12,28] for instance). More precisely, this follows from such a theorem on manifolds with corners along a hypersurface (see [4] and also [27,24]). Consider three manifolds…”

Section: Similar Tomentioning

confidence: 95%

On Hawking mass and Bartnik mass of CMC surfaces

Miao

Wang

Xie

2018

Preprint

View full text Add to dashboard Cite

Given a constant mean curvature surface that bounds a compact manifold with nonnegative scalar curvature, we obtain intrinsic conditions on the surface that guarantee the positivity of its Hawking mass. We also obtain estimates of the Bartnik mass of such surfaces, without assumptions on the integral of the squared mean curvature. If the ambient manifold has negative scalar curvature, our method also applies and yields estimates on the hyperbolic Bartnik mass of these surfaces.

show abstract

A Positive Mass Theorem on Asymptotically Hyperbolic Manifolds with Corners along a Hypersurface

Cited by 15 publications

References 16 publications

The Penrose Inequality for Asymptotically Locally Hyperbolic Spaces with Nonpositive Mass

The Penrose Inequality for Asymptotically Locally Hyperbolic Spaces with Nonpositive Mass

Nonexistence of NNSC-cobordism of Bartnik data

On Hawking mass and Bartnik mass of CMC surfaces

Contact Info

Product

Resources

About