Teichm\"uller harmonic map flow from cylinders

TL;DR

This paper introduces a geometric flow called Teichmüller harmonic map flow that deforms cylindrical surfaces into minimal surfaces spanning given boundaries, proving long-term existence and convergence to minimal solutions.

Contribution

It extends the Teichmüller harmonic map flow framework to cylindrical surfaces and establishes existence and convergence results for arbitrary initial data.

Findings

Solutions exist for all time regardless of initial data.

Flow converges to minimal surfaces under certain conditions.

Provides a method to find minimal cylinders or discs spanning boundary curves.

Abstract

We define a geometric flow that is designed to change surfaces of cylindrical type spanning two disjoint boundary curves into solutions of the Douglas-Plateau problem of finding minimal surfaces with given boundary curves. We prove that also in this new setting and for arbitrary initial data, solutions of the Teichm\"uller harmonic map flow exist for all times. Furthermore, for solutions for which a three-point-condition does not degenerate as $t\to\infty$, we show convergence along a sequence $t_i\to\infty$ to a critical point of the area given either by a minimal cylinder or by two minimal discs spanning the given boundary curves.