Existence and regularity of min-max anisotropic minimal hypersurfaces
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].
- Publication:
-
arXiv e-prints
- Pub Date:
- September 2024
- DOI:
- 10.48550/arXiv.2409.15232
- arXiv:
- arXiv:2409.15232
- Bibcode:
- 2024arXiv240915232D
- Keywords:
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- Mathematics - Differential Geometry;
- Mathematics - Analysis of PDEs;
- 53A10;
- 49Q05
- E-Print:
- 40 pages