Co-dimension one area-minimizing currents with C^1,α tangentially immersed boundary having Lipschitz co-oriented mean curvature

2018/05/02 by Rosales, Leobardo
#28A75 #49Q05 #49Q15 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1805.01287

Abstract

We study n-dimensional area-minimizing currents T in ℝⁿ⁺¹, with boundary ∂ T satisfying two properties: ∂ T is locally a finite sum of (n-1)-dimensional C^1,α orientable submanifolds which only meet tangentially and with same orientation, for some α∈ (0,1]; ∂ T has mean curvature =h ν_T where h is a Lipschitz scalar-valued function and ν_T is the generalized outward pointing normal of ∂ T with respect to T. We give a partial boundary regularity result for such currents T. We show that near any point x in the support of ∂ T, either the support of T has very uncontrolled structure, or the support of T near x is the finite union of orientable C^1,α hypersurfaces-with-boundary with disjoint interiors and common boundary points only along the support of ∂ T.

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