Minimal surfaces near Hardt-Simon foliations

2024/10/30 by Nandakumaran, Vishnu · 1 citation
#49Q05 #53A10 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2410.22795

Abstract

Caffarelli-Hardt-Simon used the minimal surface equation on the Simons cone C(S³× S³) to generate newer examples of minimal hypersurfaces with isolated singularities. Hardt-Simon proved that every area-minimizing quadratic cone C having only an isolated singularity can be approximated by a unique foliation of \mathbb Rⁿ⁺¹ by smooth, area-minimizing hypersurfaces asymptotic to \mathcal C. This paper uses methods similar to Caffareli-Hardt-Simon to solve the minimal surface equation for the Hardt-Simon surfaces in the sphere for some boundary values. We use gluing methods to construct minimal surfaces over Hardt-Simon surfaces and near quadratic cones.

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