Uniqueness of Cylindrical Tangent Cones $C_{p,q} \times \mathbb{R}$

TL;DR

This paper proves the uniqueness of a specific cylindrical tangent cone for area-minimizing hypersurfaces in nine-dimensional space, extending previous results to a new dimension.

Contribution

It establishes the uniqueness of the tangent cone $C( ext{S}^2 imes ext{S}^4) imes ext{R}$ in $ ext{R}^9$, completing the classification for cones of the form $C_{p,q} imes ext{R}$.

Findings

Uniqueness of the tangent cone $C( ext{S}^2 imes ext{S}^4) imes ext{R}$ in $ ext{R}^9$.

Extension of previous uniqueness results to the nine-dimensional case.

Completes the classification of tangent cones of the form $C_{p,q} imes ext{R}$.

Abstract

We show the uniqueness of the cylindrical tangent cone $C(\mathbb{S}^2 \times \mathbb{S}^4) \times \mathbb{R}$ for area-minimizing hypersurfaces in $\mathbb{R}^9$, completing the uniqueness of all tangent cones of the form $C_{p,q} \times \mathbb{R}$ proved by Simon for dimensions at least 10 and Sz\'ekelyhidi for the Simons cone.