Area minimizing surfaces in homotopy classes in metric spaces
Differential Geometry
2021-01-01 v1 Analysis of PDEs
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.
Cite
@article{arxiv.2012.15530,
title = {Area minimizing surfaces in homotopy classes in metric spaces},
author = {Elefterios Soultanis and Stefan Wenger},
journal= {arXiv preprint arXiv:2012.15530},
year = {2021}
}