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arXiv · 2409.15232

Existence and regularity of min-max anisotropic minimal hypersurfaces

Guido De Philippis·Antonio De Rosa·Yangyang Li
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Abstract

In any closed smooth Riemannian manifold of dimension at least three, we use the min-max construction to find anisotropic minimal hyper-surfaces with respect to elliptic integrands, with a singular set of codimension~$2$ vanishing Hausdorff measure. In particular, in a closed $3$-manifold, we obtain a smooth anisotropic minimal surface. The critical step is to obtain a uniform upper bound for density ratios in the anisotropic min-max construction. This confirms a conjecture by Allard [Invent. Math., 1983].

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Guido De Philippis, Antonio De Rosa, Yangyang Li. 2024-09-23. Existence and regularity of min-max anisotropic minimal hypersurfaces. https://arxiv.org/abs/2409.15232

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