Conjectures on Convergence and Scalar Curvature

Sormani, Christina; Curvature, Participants at the IAS Emerging Topics Workshop on Scalar; Convergence,

doi:10.48550/arxiv.2103.10093

Cited by 4 publications

(4 citation statements)

References 112 publications

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“…The positive mass stability conjecture asserts an affirmative answer to this question, although the precise statement depends on the topology used to define the closeness. The conjecture was first stated 20 years ago in terms of Gromov-Hausdorff distance by Huisken-Ilmanen in [23, page 430], and more recently in terms of intrinsic flat distance by the third author and Sormani in [27, page 216] (see also [41,Conjecture 10.1]). Definition 1.1.…”

Section: Introductionmentioning

confidence: 99%

Stability of the positive mass theorem under Ricci curvature lower bounds

Kazaras¹,

Khuri²,

Liu³

2021

Preprint

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We establish Gromov-Hausdorff stability of the Riemannian positive mass theorem under the assumption of a Ricci curvature lower bound. More precisely, consider a class of orientable complete uniformly asymptotically flat Riemannian 3-manifolds with nonnegative scalar curvature, vanishing second homology, and a uniform lower bound on Ricci curvature. We prove that if a sequence of such manifolds has ADM mass approaching zero, then it must converge to Euclidean 3-space in the pointed Gromov-Hausdorff sense. In particular, this confirms Huisken and Ilmanen's conjecture on stability of the positive mass theorem under the assumptions described above. The proof is based on the harmonic level set approach to proving the positive mass theorem, combined with techniques used in the proof of Cheeger and Colding's almost splitting theorem. Furthermore, we show that the same results hold under a more general lower bound on scalar curvature.

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Section: Introductionmentioning

confidence: 99%

Stability of the positive mass theorem under Ricci curvature lower bounds

Kazaras¹,

Khuri²,

Liu³

2021

Preprint

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“…The study of scalar curvature has a long history with many significant results. The recent lecture notes [10] and the survey paper [22] are good sources for the state of the affairs and references.…”

Section: Introductionmentioning

confidence: 99%

Comparison theorems for three-dimensional manifolds with scalar curvature bound

Munteanu¹,

Wang²

2021

Preprint

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Two sharp comparison results are derived for three-dimensional complete noncompact manifolds with scalar curvature bounded from below. The first one concerns the Green's function. When the scalar curvature is nonnegative, it states that the rate of decay of an energy quantity over the level set is strictly less than that of the Euclidean space unless the manifold itself is isometric to the Euclidean space. The result is in turn converted into a sharp area comparison for the level set of the Green's function when in addition the Ricci curvature of the manifold is assumed to be asymptotically nonnegative at infinity. The second result provides a sharp upper bound of the bottom spectrum in terms of the scalar curvature lower bound, in contrast to the classical result of Cheng which involves a Ricci curvature lower bound.

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“…See also [15] for the works on stability in term of different topology. We refer interested readers to the survey article [24] of Sormani for detailed exposition.…”

Section: Introductionmentioning

confidence: 99%