Plateau's problem for integral currents in locally non-compact metric spaces

TL;DR

This paper extends the theory of mass-minimizing integral currents to locally non-compact metric spaces, including dual Banach, injective, and Hadamard spaces, with new compactness and existence results.

Contribution

It proves existence of mass-minimizing integral currents with non-compact boundaries in various non-linear and non-compact metric spaces, generalizing prior results.

Findings

Existence of mass-minimizing integral currents in dual Banach spaces.

Weak*-compactness theorem for integral currents in dual spaces.

Generalization of previous results to broader classes of metric spaces.

Abstract

The purpose of this article is to prove existence of mass minimizing integral currents with prescribed possibly non-compact boundary in all dual Banach spaces and furthermore in certain spaces without linear structure, such as injective metric spaces and Hadamard spaces. We furthermore prove a weak$^*$-compactness theorem for integral currents in dual spaces of separable Banach spaces. Our theorems generalize results of Ambrosio-Kirchheim, Lang, the author, and recent results of Ambrosio-Schmidt.