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Uniqueness of Regular Tangent Cones for Immersed Stable Hypersurfaces

Nick Edelen, Paul Minter

Abstract

We establish uniqueness and regularity results for tangent cones (at a point or at infinity) with isolated singularities arising from a given immersed stable minimal hypersurface with suitably small (non-immersed) singular set. In particular, our results allow the tangent cone to occur with any integer multiplicity.

Uniqueness of Regular Tangent Cones for Immersed Stable Hypersurfaces

Abstract

We establish uniqueness and regularity results for tangent cones (at a point or at infinity) with isolated singularities arising from a given immersed stable minimal hypersurface with suitably small (non-immersed) singular set. In particular, our results allow the tangent cone to occur with any integer multiplicity.
Paper Structure (9 sections, 10 theorems, 92 equations)

This paper contains 9 sections, 10 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Organization
  3. Acknowledgements
  4. Preliminaries and Notation
  5. Multi-Valued Functions
  6. Estimates
  7. Proof of Theorem \ref{['thm:dini']} & \ref{['thm:main-1']}
  8. Proof of Theorem \ref{['thm:main-2']}
  9. Proof of Theorem \ref{['thm:main-3']}

Key Result

Theorem 1.1

Let $M$ be an immersed stable minimal hypersurface in $B_1^{n+1}(0)\subset\mathbb{R}^{n+1}$ (resp. $\mathbb{R}^{n+1}\setminus B_1^{n+1}(0)$) with $\mathcal{H}^{n-2}(\textnormal{sing}(M))=0$. Suppose there is a sequence $r_i\to 0$ (resp. $r_i\to \infty$) such that $r_i^{-1}M\rightharpoonup q|\mathbf{

Theorems & Definitions (26)

  • Theorem 1.1
  • Remark
  • Theorem 1.2
  • Remark
  • Theorem 1.3
  • Remark
  • Remark
  • Theorem 1.4: Dini-type a priori estimate
  • Remark 1.5
  • Lemma 2.1
  • ...and 16 more