Smooth compactness of 𝑓-minimal hypersurfaces with bounded 𝑓-index

Barbosa, Ezequiel; Sharp, Benjamin; Wei, Yong

doi:10.1090/proc/13628

Cited by 6 publications

(8 citation statements)

References 41 publications

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“…Namely for every genus g, if there exists a self shrinker of bounded entropy of that genus there is a self shrinker of least entropy of that genus. In higher dimensions, up to n = 6, Barbossa, Sharp and Wei [2] show that Colding and Minicozzi's result was true assuming an additional index bound. This note is entirely concerned with the codimension 1 case, but we also mention that in higher codimension Chen and Ma showed in [8] some compactness results for Lagrangian self shrinkers in C 2 (appropriately modifying the definition of self shrinker above).…”

Section: Introductionmentioning

confidence: 93%

Compactness and finiteness theorems for rotationally symmetric self shrinkers

Mramor¹

2020

Preprint

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In this note we first show a compactness theorem for rotationally symmetric self shrinkers of entropy less than 2, concluding that there are entropy minimizing self shrinkers diffeomorphic to S 1 × S n−1 for each n ≥ 2 in the class of rotationally symmetric self shrinkers. Assuming extra symmetry, namely that the profile curve is convex, we remove the entropy assumption. Supposing the profile curve is additionally reflection symmetric we show there are only finitely many such shrinkers up to rigid motion.

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Section: Introductionmentioning

confidence: 93%

Compactness and finiteness theorems for rotationally symmetric self shrinkers

Mramor¹

2020

Preprint

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“…(2) implies (1): We argue here as in [13,Proposition 3.2]. Let u be the positive smooth function given by item (2).…”

Section: Gradient Schrödinger Operatorsmentioning

confidence: 99%

“…This fundamental analogy with constant mean curvature hypersurfaces took the attention on the field (cf. [2,3,7,9,27,35] and reference therein).…”

Section: Introductionmentioning

confidence: 99%

Gradient Schrödinger operators, manifolds with density and applications

Espinar

2017

Journal of Mathematical Analysis and Applications

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The aim of this paper is twofold. On the one hand, the study of gradient Schrödinger operators on manifolds with density φ. We classify the space of solutions when the underlying manifold is φ−parabolic. As an application, we extend the Naber-Yau Liouville Theorem, and we will prove that a complete manifold with density is φ−parabolic if, and only if, it has finite φ−capacity. Moreover, we show that the linear space given by the kernel of a nonnegative gradient Schrödinger operators is one dimensional provided there exists a bounded function on it and the underlying manifold is φ−parabolic.On the other hand, the topological and geometric classification of complete weighted H φ −stable hypersurfaces immersed in a manifold with density (N , g, φ) satisfying a lower bound on its Bakry-Émery-Ricci tensor. Also, we classify weighted stable surfaces in a threemanifold with density whose Perelman scalar curvature, in short, P-scalar curvature, satisfies R ∞ φ + |∇φ| 2 4 ≥ 0. Here, the P-scalar curvature is defined as R ∞ φ = R − 2∆ g φ − |∇ g φ| 2 , being R the scalar curvature of (N , g).Finally, we discuss the relationship of manifolds with density, Mean Curvature Flow (MCF), Ricci Flow and Optimal Transportation Theory. We obtain classification results for stable self-similiar solutions to the MCF, and also for stable translating solitons to the MCF (as far as we know, the first classification result). Moreover, for gradient Ricci solitons, we recover the Hamilton-Ivey-Perelman classification assuming only a L 2 −type bound on the scalar curvature and a inequality between the scalar curvature and the Ricci tensor. We also classify gradient Ricci solitons when the scalar curvature does not change sign. We finish by classifying critical transportation plans for the Boltzman entropy on φ−parabolic manifolds. In particular, we recover the case for the Gaussian measure.

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“…The idea in [17] has been used by the authors and Sharp [18] to obtain an analogous smooth compactness for the space of complete f -minimal hypersurfaces of dimension 2 ≤ n ≤ 6 and in particular the space of self-shrinkers. This motivates the natural question: Can we obtain a smooth compactness theorem for the space of free boundary minimal (or f -minimal) hypersurface with bounded index (or f -index) and bounded volume?…”

Section: Remark 11mentioning

confidence: 99%

A Compactness Theorem of the Space of Free Boundary f-Minimal Surfaces in Three-Dimensional Smooth Metric Measure Space with Boundary

Barbosa

Wei

2015

J Geom Anal

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Let (M 3 , g, e − f dμ M ) be a compact three-dimensional smooth metric measure space with nonempty boundary. Suppose that M has nonnegative Bakry-Émery Ricci curvature and the boundary ∂ M is strictly f -mean convex. We prove that there exists a properly embedded smooth f -minimal surface in M with free boundary ∂ on ∂ M. If we further assume that the boundary ∂ M is strictly convex, then we prove that M 3 is diffeomorphic to the 3-ball B 3 , and a compactness theorem for the space of properly embedded f -minimal surfaces with free boundary in such (M 3 , g, e − f dμ M ), when the topology of these f -minimal surfaces is fixed.

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Smooth compactness of 𝑓-minimal hypersurfaces with bounded 𝑓-index

Cited by 6 publications

References 41 publications

Compactness and finiteness theorems for rotationally symmetric self shrinkers

Compactness and finiteness theorems for rotationally symmetric self shrinkers

Gradient Schrödinger operators, manifolds with density and applications

A Compactness Theorem of the Space of Free Boundary f-Minimal Surfaces in Three-Dimensional Smooth Metric Measure Space with Boundary

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