Minimal hypersurfaces with cylindrical tangent cones

2021/07/30 by Székelyhidi, Gábor · 2 citations
#53A10 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2107.14786

Abstract

First we construct minimal hypersurfaces M\subsetRⁿ⁺¹ in a neighborhood of the origin, with an isolated singularity but cylindrical tangent cone C× R, for any strictly minimizing strictly stable cone C in Rⁿ. We show that many of these hypersurfaces are area minimizing. Next, we prove a strong unique continuation result for minimal hypersurfaces V with such a cylindrical tangent cone, stating that if the blowups of V centered at the origin approach C× R at infinite order, then V = C\timesR in a neighborhood of the origin. Using this we show that for quadratic cones C = C(S^p × S^q), in dimensions n > 8, all O(p+1) × O(q+1)-invariant minimal hypersurfaces with tangent cone C× R at the origin are graphs over one of the surfaces that we constructed. In particular such an invariant minimal hypersurface is either equal to C× R or has an isolated singularity at the origin.

Cited by

• Minimal surfaces and comparison geometry
• Uniqueness in the Plateau problem near Quadratic cones

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