The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals

Guido De Philippis; Antonio De Rosa; Jonas Hirsch

doi:10.3934/dcds.2019243

The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals

Guido De Philippis, Antonio De Rosa, Jonas Hirsch

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

In this paper we investigate the “area blow-up” set of a sequence of smooth co-dimension one manifolds whose first variation with respect to an anisotropic integral is bounded. Following the ideas introduced by White in [12], we show that this set has bounded (anisotropic) mean curvature in the viscosity sense. In particular, this allows to show that the set is empty in a variety of situations. As a consequence, we show boundary curvature estimates for two dimensional stable anisotropic minimal surfaces, extending the results of [10].

Original language	English (US)
Pages (from-to)	7031-7056
Number of pages	26
Journal	Discrete and Continuous Dynamical Systems- Series A
Volume	39
Issue number	12
DOIs	https://doi.org/10.3934/dcds.2019243
State	Published - 2019

Keywords

Anisotropic energies
Curvature estimates
Geometric analysis
Minimal surfaces
Viscosity solutions

ASJC Scopus subject areas

Analysis
Discrete Mathematics and Combinatorics
Applied Mathematics

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10.3934/dcds.2019243

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abstract = "In this paper we investigate the “area blow-up” set of a sequence of smooth co-dimension one manifolds whose first variation with respect to an anisotropic integral is bounded. Following the ideas introduced by White in [12], we show that this set has bounded (anisotropic) mean curvature in the viscosity sense. In particular, this allows to show that the set is empty in a variety of situations. As a consequence, we show boundary curvature estimates for two dimensional stable anisotropic minimal surfaces, extending the results of [10].",

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AU - De Philippis, Guido

AU - De Rosa, Antonio

AU - Hirsch, Jonas

PY - 2019

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N2 - In this paper we investigate the “area blow-up” set of a sequence of smooth co-dimension one manifolds whose first variation with respect to an anisotropic integral is bounded. Following the ideas introduced by White in [12], we show that this set has bounded (anisotropic) mean curvature in the viscosity sense. In particular, this allows to show that the set is empty in a variety of situations. As a consequence, we show boundary curvature estimates for two dimensional stable anisotropic minimal surfaces, extending the results of [10].

AB - In this paper we investigate the “area blow-up” set of a sequence of smooth co-dimension one manifolds whose first variation with respect to an anisotropic integral is bounded. Following the ideas introduced by White in [12], we show that this set has bounded (anisotropic) mean curvature in the viscosity sense. In particular, this allows to show that the set is empty in a variety of situations. As a consequence, we show boundary curvature estimates for two dimensional stable anisotropic minimal surfaces, extending the results of [10].

KW - Anisotropic energies

KW - Curvature estimates

KW - Geometric analysis

KW - Minimal surfaces

KW - Viscosity solutions

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The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals

Abstract

Keywords

ASJC Scopus subject areas

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