Hyperbolic Unfoldings of Minimal Hypersurfaces

IF 0.9 3区数学 Q2 MATHEMATICS

Analysis and Geometry in Metric Spaces Pub Date : 2018-05-06 DOI:10.1515/agms-2018-0006

J. Lohkamp

引用次数: 8

Abstract

Abstract We study the intrinsic geometry of area minimizing hypersurfaces from a new point of view by relating this subject to quasiconformal geometry. Namely, for any such hypersurface H we define and construct a so-called S-structure. This new and natural concept reveals some unexpected geometric and analytic properties of H and its singularity set Ʃ. Moreover, it can be used to prove the existence of hyperbolic unfoldings of H\Ʃ. These are canonical conformal deformations of H\Ʃ into complete Gromov hyperbolic spaces of bounded geometry with Gromov boundary homeomorphic to Ʃ. These new concepts and results naturally extend to the larger class of almost minimizers.

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最小超曲面的双曲展开

摘要将面积最小化超曲面与拟共形几何联系起来，从一个新的角度研究了面积最小化超曲面的内在几何问题。也就是说，对于任何这样的超曲面H，我们定义并构造一个所谓的s结构。这个新的自然概念揭示了H及其奇异集Ʃ的一些意想不到的几何和解析性质。此外，它还可以用来证明H\Ʃ双曲展开的存在性。这些是H\Ʃ在具有格罗莫夫边界同纯于Ʃ的有界几何的完全格罗莫夫双曲空间中的正则共形变形。这些新的概念和结果自然延伸到更大的一类几乎最小化。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Analysis and Geometry in Metric Spaces Mathematics-Geometry and Topology

CiteScore

1.80

自引率

0.00%

发文量

审稿时长

16 weeks

期刊介绍： Analysis and Geometry in Metric Spaces is an open access electronic journal that publishes cutting-edge research on analytical and geometrical problems in metric spaces and applications. We strive to present a forum where all aspects of these problems can be discussed. AGMS is devoted to the publication of results on these and related topics: Geometric inequalities in metric spaces, Geometric measure theory and variational problems in metric spaces, Analytic and geometric problems in metric measure spaces, probability spaces, and manifolds with density, Analytic and geometric problems in sub-riemannian manifolds, Carnot groups, and pseudo-hermitian manifolds. Geometric control theory, Curvature in metric and length spaces, Geometric group theory, Harmonic Analysis. Potential theory, Mass transportation problems, Quasiconformal and quasiregular mappings. Quasiconformal geometry, PDEs associated to analytic and geometric problems in metric spaces.