Energy and area minimizers in metric spaces

TL;DR

This paper demonstrates that in proper metric spaces, solutions to the classical Plateau problem can be obtained by energy minimization with an appropriate convex geometry-based area definition, leading to regularity results.

Contribution

It introduces a new area definition in metric spaces, proves its quasi-convexity, and establishes regularity of energy minimizers under quadratic isoperimetric conditions.

Findings

Existence of Plateau problem solutions via energy minimization.

Quasi-convexity of the new area definition.

Improved Hölder regularity of area-minimizing discs.

Abstract

We show that in the setting of proper metric spaces one obtains a solution of the classical two-dimensional Plateau problem by minimizing the energy, as in the classical case, once a definition of area (in the sense of convex geometry) has been chosen appropriately. We prove the quasi-convexity of this new definition of area. Under the assumption of a quadratic isoperimetric inequality we establish regularity results for energy minimizers and improve Hoelder exponents of some area-minimizing discs.