Minimal hypersurfaces with cylindrical tangent cones

TL;DR

This paper constructs and analyzes minimal hypersurfaces with cylindrical tangent cones, proving uniqueness and classification results for such hypersurfaces near singularities, especially for quadratic cones in high dimensions.

Contribution

It introduces a method to construct minimal hypersurfaces with cylindrical tangent cones and establishes a strong unique continuation principle for these hypersurfaces.

Findings

Many constructed hypersurfaces are area minimizing.

Unique continuation holds for hypersurfaces with cylindrical tangent cones.

Invariant hypersurfaces are either cones or have isolated singularities.

Abstract

First we construct minimal hypersurfaces $M\subset\mathbf{R}^{n+1}$ in a neighborhood of the origin, with an isolated singularity but cylindrical tangent cone $C\times \mathbf{R}$, for any strictly minimizing strictly stable cone $C$ in $\mathbf{R}^n$. 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 $C\times \mathbf{R}$ at infinite order, then $V = C\times\mathbf{R}$ in a neighborhood of the origin. Using this we show that for quadratic cones $C = C(S^p \times S^q)$, in dimensions $n > 8$, all $O(p+1) \times O(q+1)$-invariant minimal hypersurfaces with tangent cone $C\times \mathbf{R}$ at the origin are graphs over one of the surfaces that we constructed. In particular such an invariant minimal hypersurface is either equal to $C\times \mathbf{R}$ or has an isolated singularity at the origin.