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On the Boundary Behavior of Mass-Minimizing Integral Currents

About this Title

Camillo De Lellis, Guido De Philippis, Jonas Hirsch and Annalisa Massaccesi

Publication: Memoirs of the American Mathematical Society
Publication Year: 2023; Volume 291, Number 1446
ISBNs: 978-1-4704-6695-4 (print); 978-1-4704-7679-3 (online)
DOI: https://doi.org/10.1090/memo/1446
Published electronically: November 8, 2023
Keywords: Boundary regularity, area-minimizing currents, minimal surfaces, calculus of variations, geometric measure theory

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Table of Contents

Chapters

  • 1. Introduction
  • 2. Corollaries, open problems, and plan of the paper
  • 3. Stratification and reduction to collapsed points
  • 4. Regularity for $\left (Q-{\textstyle \frac 12}\right )$ $\mathrm {Dir}$-minimizers
  • 5. First Lipschitz approximation and harmonic blow-up
  • 6. Decay of the excess and uniqueness of tangent cones
  • 7. Second Lipschitz approximation
  • 8. Center manifolds
  • 9. Monotonicity of the frequency function
  • 10. Final blow-up argument

Abstract

Let $\Sigma$ be a smooth Riemannian manifold, $\Gamma \subset \Sigma$ a smooth closed oriented submanifold of codimension higher than $2$ and $T$ an integral area-minimizing current in $\Sigma$ which bounds $\Gamma$. We prove that the set of regular points of $T$ at the boundary is dense in $\Gamma$. Prior to our theorem the existence of any regular point was not known, except for some special choice of $\Sigma$ and $\Gamma$. As a corollary of our theorem

  • we answer to a question in Almgren’s Almgren’s big regularity paper from 2000 showing that, if $\Gamma$ is connected, then $T$ has at least one point $p$ of multiplicity $\frac {1}{2}$, namely there is a neighborhood of the point $p$ where $T$ is a classical submanifold with boundary $\Gamma$;
  • we generalize Almgren’s connectivity theorem showing that the support of $T$ is always connected if $\Gamma$ is connected;
  • we conclude a structural result on $T$ when $\Gamma$ consists of more than one connected component, generalizing a previous theorem proved by Hardt and Simon in 1979 when $\Sigma = \mathbb R^{m+1}$ and $T$ is $m$-dimensional.

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