The Area Blow Up set for bounded mean curvature submanifolds with\n respect to elliptic surface energy functionals

2019/01/11 by Guido De Philippis, A. De Rosa, De Philippis, Guido +3 · 2 citations
Computer Science · Mathematics · #35D40 #49Q05 #53A10 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1901.03514

openalex publication_date 2019/01/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we investigate the "area blow-up" set of a sequence of smooth\nco-dimension one manifolds whose first variation with respect to an anisotropic\nintegral is bounded. Following the ideas introduced by White in (J.\nDifferential Geom., 2016), we show that this set has bounded (anisotropic) mean\ncurvature in the viscosity sense. In particular, this allows to show that the\nset is empty in a variety of situations. As a consequence, we show boundary\ncurvature estimates for two dimensional stable anisotropic minimal surfaces,\nextending the results of (Invent. Math., 1987).\n

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