Corpus ID: 203610459

Regularity of minimal surfaces near quadratic cones

@article{Edelen2019RegularityOM,
title={Regularity of minimal surfaces near quadratic cones},
author={Nick Edelen and Luca Spolaor},
journal={arXiv: Differential Geometry},
year={2019}
}
• Published 1 October 2019
• Mathematics
• arXiv: Differential Geometry
Hardt-Simon proved that every area-minimizing hypercone $\mathbf{C}$ having only an isolated singularity fits into a foliation of $\mathbb{R}^{n+1}$ by smooth, area-minimizing hypersurfaces asymptotic to $\mathbf{C}$. In this paper we prove that if a stationary $n$-varifold $M$ in the unit ball $B_1 \subset \mathbb{R}^{n+1}$ lies sufficiently close to a minimizing quadratic cone (for example, the Simons' cone $\mathbf{C}^{3,3}$), then $\mathrm{spt} M \cap B_{1/2}$ is a $C^{1,\alpha… Expand 4 Citations Uniqueness of certain cylindrical tangent cones We show that the cylindrical tangent cone$C\times \mathbf{R}$for an area-minimizing hypersurface is unique, where$C$is the Simons cone$C_S= C(S^3\times S^3)$. Previously Simon proved aExpand Deformations of Singular Minimal Hypersurfaces I, Isolated Singularities. Locally stable minimal hypersurface could have singularities in dimension$\geq 7$in general, locally modeled on stable and area-minimizing cones in the Euclidean spaces. In this paper, we presentExpand Minimal hypersurfaces with cylindrical tangent cones First we construct minimal hypersurfaces M ⊂ R in a neighborhood of the origin, with an isolated singularity but cylindrical tangent cone C × R, for any strictly minimizing strictly stable cone C inExpand Regularity of free boundary minimal surfaces in locally polyhedral domains • Mathematics • 2020 We prove an Allard-type regularity theorem for free-boundary minimal surfaces in Lipschitz domains locally modelled on convex polyhedra. We show that if such a minimal surface is sufficiently closeExpand References SHOWING 1-10 OF 13 REFERENCES The singular set of minimal surfaces near polyhedral cones • Mathematics • 2017 We adapt the method of Simon [JDG '93] to prove a$C^{1,\alpha}$-regularity theorem for minimal varifolds which resemble a cone$\bf{C}_0^2\$ over an equiangular geodesic net. For varifold classesExpand
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It is known ([SL1]) that if a minimal submanifold M has a tangent cone C at a singular point p, then C is the unique tangent cone of M at p, and M approaches C asymptotically in the appropriateExpand
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Suppose V is an m + 1-dimensional minimal surface in R" containing 0 as an isolated singular point. What can one say about the structure of V near O? This question initiated the present investigationExpand
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The question of what can be said about the structure of the singular set of minimal surfaces and the extrema of other geometric variational problems has remained largely open. Indeed, for minimalExpand
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Suppose M is a smooth m dimensional Riemannian manifold and k is a positive integer not exceeding m. Our purpose is to study the first variation of the k dimensional area integrand in M. Our mainExpand
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For n≥3, there exists an embedded minimal hypersurface in Rn+1 which has an isolated singularity but which is not a cone. Each example constructed here is asymptotic to a given, completely arbitrary,Expand
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