Inverse spectral positivity for surfaces

Bérard, Pierre; Castillon, Philippe

doi:10.4171/rmi/813

Cited by 9 publications

(11 citation statements)

References 22 publications

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“…The result in the following proposition was established under the additional hypothesis that the Gauss curvature of the immersion be integrable in [32,Theorem 3 (ii)] and left as an open problem in [32,Remark 2]. Solutions have been proposed in [65,50,6,58]. 4 Here we present a short proof based on a result by D. Fischer-Colbrie.…”

Section: Appendix C Rigidity Of Stable Minimal Cylindersmentioning

confidence: 85%

Effective versions of the positive mass theorem

2016

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The study of stable minimal surfaces in Riemannian 3-manifolds (M, g) with non-negative scalar curvature has a rich history. In this paper, we prove rigidity of such surfaces when (M, g) is asymptotically flat and has horizon boundary. As a consequence, we obtain an effective version of the positive mass theorem in terms of isoperimetric or, more generally, closed volume-preserving stable CMC surfaces that is appealing from both a physical and a purely geometric point of view. We also include a proof of the following conjecture of Schoen: An asymptotically flat Riemannian 3-manifold with non-negative scalar curvature that contains an unbounded area-minimizing surface is isometric to flat R 3 .

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Section: Appendix C Rigidity Of Stable Minimal Cylindersmentioning

confidence: 85%

Effective versions of the positive mass theorem

2016

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“…Since [9], the relation between the non-negativity of operators of the type −Δ + aK + V (a ∈ R + , V ∈ L 1 loc (M )) on a surface M and the conformal type of M has been the subject of increasing interest. A beautiful and general result appeared in [2], and we refer to this paper also for an up-to-date account on the problem. …”

Section: ) Yieldsmentioning

confidence: 93%

A note on Killing fields and CMC hypersurfaces

Mari

Mastrolia

Rigoli

2015

Journal of Mathematical Analysis and Applications

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In this note we give some sufficient conditions for a CMC-hypersurface in a Riemannian manifold N to be invariant under the 1-parameter group of isometries generated by a Killing field on N . Our main result improves on previous ones by hinges on a new, simple existence theorem for a first zero of solutions of an ODE naturally associated to the problem. This theorem implies some classical oscillation criteria of W. Ambrose and R. Moore. Extension to constant higher-order mean curvature hypersurfaces are also presented.

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“…We refer to [23,17,3,20] as well as Appendix C of [8] for discussions and proofs of the following refinement of the latter alternative in Lemma 2.2.…”

Section: Toolsmentioning

confidence: 99%

“…We may repeat the above argument starting with any of the surfacesΣ(r) for r > 0 sufficiently small, using that they are homologically* area-minimizing in (M ,ĝ). 3 A continuity argument then gives that (M ,ĝ) is either standard S 1 × R × [0, ∞) or standard S 1 × R × [0, a] for some a > 0. Lemma 3.1.…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

A Splitting Theorem for Scalar Curvature

Chodosh

Eichmair

Moraru

2018

Comm Pure Appl Math

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We dedicate this paper to Gregory Galloway on the occasion of his 70 th birthday. AbstractWe show that a Riemannian 3-manifold with nonnegative scalar curvature is flat if it contains an area-minimizing cylinder. This scalar-curvature analogue of the classical splitting theorem of J. Cheeger and D. Gromoll (1971) was conjectured by D. Fischer-Colbrie and R. Schoen (1980) and by M. Cai and G. Galloway (2000).Let .M; g/ be a connected, orientable, complete Riemannian 3-manifold with nonnegative scalar curvature. D. Fischer-Colbrie and R. Schoen show in [10] that a connected, orientable, complete stable minimal immersion into .M; g/ is conformal to a plane, a sphere, a torus, or a cylinder. They conjecture that .M; g/ is flat if the immersion is conformal to the cylinder; cf. remark 4 in [10]. M. Cai and G. Galloway point out a counterexample obtained from flattening standard R S 2 near R fgreat circleg in their concluding remark in [7]. They ask if the conjecture holds under the additional assumption that the immersion be "suitably" area-minimizing. In this paper, we prove the following result: THEOREM 1.1. Let .M; g/ be a connected, orientable, complete Riemannian 3manifold with nonnegative scalar curvature. Assume that .M; g/ contains a properly embedded surface S M that is both homeomorphic to the cylinder and absolutely area-minimizing. Then .M; g/ is flat. In fact, a cover of .M; g/ is isometric to standard S 1 R 2 upon scaling.Note that this result is in satisfying analogy with the classical splitting theorem of J. Cheeger and D. Gromoll [9] in dimension 3, where scalar curvature takes the

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Inverse spectral positivity for surfaces

Cited by 9 publications

References 22 publications

Effective versions of the positive mass theorem

Effective versions of the positive mass theorem

A note on Killing fields and CMC hypersurfaces

A Splitting Theorem for Scalar Curvature

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