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arXiv · 2609.20459

Regularity of varifolds with bounded anisotropic first variation

Antonio De Rosa·Benjy Firester·Raphael Tsiamis
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Abstract

We prove an $\varepsilon$-regularity theorem for $m$-varifolds with mean curvature in $L^p$, $p>m$, with respect to an anisotropic integrand satisfying a Michael-Simon inequality and the quadratic exposed condition: near sufficiently flat density-one points, such varifolds are representable as $C^{1,α}$ graphs. Combined with the recent proof of the anisotropic Michael-Simon inequality, this establishes an anisotropic Allard regularity theorem in arbitrary codimension for a large class of anisotropic integrands, including those close to the area functional. We also exhibit the first examples of anisotropies satisfying both the uniform scalar atomic condition and the Michael-Simon inequality, that are not close to any ellipsoidal norm. These include the $\ell^q$ norms in every dimension and codimension, for explicit ranges of $q$, and a new class of axisymmetric anisotropies.

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Antonio De Rosa, Benjy Firester, Raphael Tsiamis. 2026-09-17. Regularity of varifolds with bounded anisotropic first variation. https://arxiv.org/abs/2609.20459

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