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Area minimizing discs in locally non-compact metric spaces

Chang-Yu Guo, Stefan Wenger

Abstract

We solve the classical problem of Plateau in every metric space which is $1$-complemented in an ultra-completion of itself. This includes all proper metric spaces as well as many locally non-compact metric spaces, in particular, all dual Banach spaces, some non-dual Banach spaces such as $L^1$, all Hadamard spaces, and many more. Our results generalize corresponding results of Lytchak and the second author from the setting of proper metric spaces to that of locally non-compact ones. We furthermore solve the Dirichlet problem in the same class of spaces. The main new ingredient in our proofs is a suitable generalization of the Rellich-Kondrachov compactness theorem, from which we deduce a result about ultra-limits of sequences of Sobolev maps.

Area minimizing discs in locally non-compact metric spaces

Abstract

We solve the classical problem of Plateau in every metric space which is -complemented in an ultra-completion of itself. This includes all proper metric spaces as well as many locally non-compact metric spaces, in particular, all dual Banach spaces, some non-dual Banach spaces such as , all Hadamard spaces, and many more. Our results generalize corresponding results of Lytchak and the second author from the setting of proper metric spaces to that of locally non-compact ones. We furthermore solve the Dirichlet problem in the same class of spaces. The main new ingredient in our proofs is a suitable generalization of the Rellich-Kondrachov compactness theorem, from which we deduce a result about ultra-limits of sequences of Sobolev maps.

Paper Structure

This paper contains 9 sections, 17 theorems, 61 equations.

Table of Contents

  1. Introduction and statement of main results
  2. Preliminaries
  3. Basic notation
  4. Ultra-completions of metric spaces
  5. Metric space valued Sobolev maps
  6. Generalized Rellich-Kondrachov theorem
  7. Ultra-limits of subsequences of Sobolev maps
  8. Area and energy minimizers with prescribed Jordan boundary
  9. Dirichlet's problem

Key Result

Theorem 1.2

Let $X$ be a complete metric space and $\Gamma$ a Jordan curve in $X$ such that $\Lambda(\Gamma, X)\not= \emptyset$. If $X$ is $1$-complemented in some ultra-completion of $X$ then there exists $u\in \Lambda(\Gamma, X)$ such that and $u$ is $\sqrt{2}$-quasiconformal.

Theorems & Definitions (33)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 23 more