Area minimizing surfaces in homotopy classes in metric spaces

TL;DR

This paper develops a framework for studying area minimizing surfaces within specific homotopy classes in metric spaces, establishing existence, regularity, and homotopy equivalence results under certain geometric conditions.

Contribution

It introduces a new notion of relative 1-homotopy type for Sobolev maps and proves existence and regularity of area minimizing surfaces in this setting, generalizing previous results.

Findings

Existence of area minimizing surfaces in relative 1-homotopy classes.

Local Hölder regularity of these surfaces.

Homotopy equivalence results in spaces with trivial second homotopy group.

Abstract

We introduce and study a notion of relative 1-homotopy type for Sobolev maps from a surface to a metric space spanning a given collection of Jordan curves. We use this to establish the existence and local H\"older regularity of area minimizing surfaces in a given relative 1-homotopy class in proper geodesic metric spaces admitting a local quadratic isoperimetric inequality. If the underlying space has trivial second homotopy group then relatively 1-homotopic maps are relatively homotopic. We also obtain an analog for closed surfaces in a given 1-homotopy class. Our theorems generalize and strengthen results of Lemaire, Jost, Schoen-Yau, and Sacks-Uhlenbeck.