Isoperimetric inequalities for submanifolds. Jellett-Minkowski's formula revisited

Gimeno, Vicent

doi:10.1112/plms/pdu053

Cited by 9 publications

(8 citation statements)

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“…Then, by using Corollary 2.7 of [Gim14], Σ has positive Cheeger constant h(Σ) > 0, in particular the end of revolution E has also positive Cheeger constant h(E) > 0, and therefore E has positive fundamental tone λ * (E) which implies that E is non-parabolic (see [Gri99]) in contradiction with Theorem A.…”

Section: Examples Of Applicationmentioning

confidence: 52%

“…From the conformally point of view the Brownian motion of any complete bounded minimal surface in R 3 is transcient (non-recurrent) (see [BM07] for instance). Moreover, the Brownian movement of a submanifold is transcient (see [Gim14]) if the submanifold admits a complete immersion within a geodesic ball of radius R with mean curvature vector field H bounded by H < 1 R Taking into account that by Theorem A any end of revolution in R 3 is a parabolic end we can state Corollary 9.1. Let Σ be a surface isometrically immersed into a geodesic ball B R ⊂ R 3 .…”

Section: Examples Of Applicationmentioning

confidence: 99%

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Conformal type of ends of revolution in space forms of constant sectional curvature

Gimeno

Gozalbo

2015

Ann Glob Anal Geom

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ABSTRACT. In this paper we consider the conformal type (parabolicity or non-parabolicity) of complete ends of revolution immersed in simply connected space forms of constant sectional curvature. We show that any complete end of revolution in the 3-dimensional Euclidean space or in the 3-dimensional sphere is parabolic. In the case of ends of revolution in the hyperbolic 3-dimensional space, we find sufficient conditions to attain parabolicity for complete ends of revolution using their relative position to the complete flat surfaces of revolution. This paper is concerned with the study of the conformal type (parabolicity or nonparabolicity) of complete ends of revolution immersed in the 3-dimensional Euclidean space R 3 , in the 3-dimensional hyperbolic space H 3 , or in the 3-dimensional sphere S 3 . Let us denote by M 3 (κ) the simply connected space form of constant sectional curvature κ ∈ R. Hence,is a complete end of revolution if there exists a geodesic in M 3 (κ) such that the end is invariant by the group of rotations of M 3 (κ) that leaves this geodesic point-wise fixed. More precisely, an end of revolution will be the rotation along a geodesic ray γ of M 3 (κ) of a generating smooth curve β : [0, ∞) → M 2 (κ) contained in a totally geodesic 2010 Mathematics Subject Classification. Primary 53C20 53C40; Secondary 53C42.

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Section: Examples Of Applicationmentioning

confidence: 52%

Section: Examples Of Applicationmentioning

confidence: 99%

Conformal type of ends of revolution in space forms of constant sectional curvature

Gimeno

Gozalbo

2015

Ann Glob Anal Geom

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“…Brendle and M. Eichmair [8] extended Brendle's result to the closed, convex, starshaped hypersurfaces with constant higher order mean curvature. See also [18] by V. Gimeno,[26] by J. Li and C. Xia, and [36] by X. Wang and Y.-K. Wang.…”

Section: Motivation and Main Resultsmentioning

confidence: 99%

Weighted Hsiung-Minkowski formulas and rigidity of umbilical hypersurfaces

Kwong

Lee

Pyo

2018

Mathematical Research Letters

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We use the weighted Hsiung-Minkowski integral formulas and Brendle's inequality to show new rigidity results. First, we prove Alexandrov type results for closed embedded hypersurfaces with radially symmetric higher order mean curvature in a large class of Riemannian warped product manifolds, including the Schwarzschild and Reissner-Nordström spaces, where the Alexandrov reflection principle is not available. Second, we prove that, in Euclidean space, the only closed immersed self-expanding solitons to the weighted generalized inverse curvature flow of codimension one are round hyperspheres.2010 Mathematics Subject Classification. 53C44, 53C42.

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“…The classical theorems of Jellett, Alexandrov, and Hopf show that round spheres posses some rigidity as CMC hypersurfaces. Jellett's Theorem in R 3 and its generalization [26,35,36,44,50], which uses Hsiung-Minkowski integral formulas [35,36], confirms that a closed, star-shaped, CMC hypersurface is round. Alexandrov used his method of moving planes to prove that an embedded, closed CMC hypersurface in R n+1 must be a round sphere.…”

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confidence: 76%