Rigidity of Area-Minimizing Free Boundary Surfaces in Mean Convex Three-Manifolds

Ambrozio, Lucas

doi:10.1007/s12220-013-9453-2

Cited by 41 publications

(59 citation statements)

References 13 publications

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“…Lastly, the fact that S q,j,α is separable is obtained as follows. Recall that, by virtue of statement (2) in Proposition 41 if w ∈ C j,α (M, N ) is γ-stationary then in fact w ∈ C q (M, N ), hence letŠ q,j,α be just the set S q,j,α endowed with the topology it inherits as a subspace of the product Γ×(C q (M, N )/ ≃), for ≃ the usual equivalence relation (quotienting by diffeomorphisms of M). Clearly, S q,j,α has a coarser topology thanŠ q,j,α so it is enough to show thatŠ q,j,α is separable.…”

Section: A Bumpy Metric Theorem For Free Boundary Minimal Hypersurfacesmentioning

confidence: 99%

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Compactness analysis for free boundary minimal hypersurfaces

2017

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We investigate compactness phenomena involving free boundary minimal hypersurfaces in Riemannian manifolds of dimension less than eight. We provide natural geometric conditions that ensure strong one-sheeted graphical subsequential convergence, discuss the limit behaviour when multi-sheeted convergence happens and derive various consequences in terms of finiteness and topological control.Mathematics Subject Classification (MSC 2010): Primary 53A10; Secondary 53C42, 49Q05.

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Section: A Bumpy Metric Theorem For Free Boundary Minimal Hypersurfacesmentioning

confidence: 99%

“…The functional Φ is C 1 (see e. g. Appendix of [39]) and its differential can be computed using the result of Proposition 17 in [2], so that…”

Section: Appendix B Proof Of Proposition 21mentioning

confidence: 99%

Compactness analysis for free boundary minimal hypersurfaces

2017

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“…Subsequent to the paper of M. Cai and G. Galloway [7], there have been many further works establishing scalar curvature rigidity results in the presence of compact area-minimizing surfaces, including [6,4,5,11,14,15,19,1,16,18]. We anticipate that the techniques developed here lead to alternative proofs of these results.…”

Section: Introductionmentioning

confidence: 82%

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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“…Recently one of inspiring work is a series of papers of Fraser-Schoen [22,23,24] about minimal hypersurfaces with free boundary in a ball and the first Steklov eigenvalue. See also [11,15,63,35,20,25,3]. Our research on the stability on CMC hypersurfaces are motivated by these results.…”

Section: Introductionmentioning

confidence: 94%

Uniqueness of stable capillary hypersurfaces in a ball

Wang

Xia

2019

Math. Ann.

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In this paper we prove that any immersed stable capillary hypersurfaces in a ball in space forms are totally umbilical. Our result also provides a proof of a conjecture proposed by Sternberg-Zumbrun in J Reine Angew Math 503 (1998), 63-85. We also prove a Heintze-Karcher-Ros type inequality for hypersurfaces with free boundary in a ball, which, together with the new Minkowski formula, yields a new proof of Alexandrov's Theorem for embedded CMC hypersurfaces in a ball with free boundary.Part of this work was done while CX was visiting the mathematical institute of Albert-Ludwigs-Universität Freiburg. He would like to thank the institute for its hospitality.

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Rigidity of Area-Minimizing Free Boundary Surfaces in Mean Convex Three-Manifolds

Cited by 41 publications

References 13 publications

Compactness analysis for free boundary minimal hypersurfaces

Compactness analysis for free boundary minimal hypersurfaces

A Splitting Theorem for Scalar Curvature

Uniqueness of stable capillary hypersurfaces in a ball

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