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  Curvature of closed subsets of Euclidean space and minimal submanifolds of arbitrary codimension

Santilli, M. (submitted). Curvature of closed subsets of Euclidean space and minimal submanifolds of arbitrary codimension.

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Item Permalink: http://hdl.handle.net/11858/00-001M-0000-002D-E1B7-B Version Permalink: http://hdl.handle.net/21.11116/0000-0002-F9CF-D
Genre: Paper

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1708.01549.pdf (Preprint), 452KB
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Santilli, Mario1, Author              
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1Geometric Measure Theory, AEI-Golm, MPI for Gravitational Physics, Max Planck Society, ou_1753352              

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Free keywords: Mathematics, Differential Geometry, math.DG,Mathematics, Analysis of PDEs, math.AP,Mathematics, Metric Geometry, math.MG,
 Abstract: A second fundamental form is introduced for arbitrary closed subsets of Euclidean space, extending the same notion introduced by J. Fu for sets of positive reach. We extend well known integral-geometric formulas to this general setting and we provide a structural result in terms of second fundamental forms of submanifolds of class $2$ that is new even for sets of positive reach. In the case of a large class of minimal submanifolds, which include viscosity solutions of the minimal surface system and rectifiable stationary varifolds of arbitrary codimension and higher multiplicities, we prove the area formula for the generalized Gauss map in terms of the discriminant of the second fundamental form and, adapting techniques from the theory of viscosity solutions of elliptic equations to our geometric setting, we conclude a natural second-order-differentiability property almost everywhere. Moreover the trace of the second fundamental form is proved to be zero for stationary integral varifolds.

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 Dates: 2017-08-042017
 Publication Status: Submitted
 Pages: 35 pages
 Publishing info: -
 Table of Contents: -
 Rev. Method: -
 Identifiers: arXiv: 1708.01549
URI: http://arxiv.org/abs/1708.01549
 Degree: -

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