Rectifiability of varifolds with locally bounded first variation with respect to anisotropic surface energies

Antonio De Rosa; Francesco Ghiraldin; Guido De Philippis

Rectifiability of varifolds with locally bounded first variation with respect to anisotropic surface energies

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arxiv 1609.01908 v3 pith:NMJQEAHT submitted 2016-09-07 math.AP

Guido De Philippis , Antonio De Rosa , Francesco Ghiraldin This is my paper

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keywords boundedfirstintegrandlocallyrectifiabilityrespectvariationanisotropic

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abstract

We extend Allard's celebrated rectifiability theorem to the setting of varifolds with locally bounded first variation with respect to an anisotropic integrand. In particular, we identify a sufficient and necessary condition on the integrand to obtain the rectifiability of every \(d\)-dimensional varifold with locally bounded first variation and positive \(d\)-dimensional density. In codimension one, this condition is shown to be equivalent to the strict convexity of the integrand with respect to the tangent plane.

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Geometry of the Atomic Condition
math.AP 2026-07 conditional novelty 8.0 of 10

Strict polyconvexity does not imply the atomic condition for k-dimensional surface energies once n−k≥3; the atomic condition itself has a convex-geometric characterization.

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