Global weak solutions of the Teichm\"uller harmonic map flow into general targets

TL;DR

This paper studies the Teichmüller harmonic map flow, providing a method to extend solutions past singularities and demonstrating that the flow decomposes any map into minimal immersions without losing topological information.

Contribution

It introduces a canonical extension method for the flow beyond finite-time singularities and proves a no-topology-loss result, enabling global solutions for general targets.

Findings

Constructed global solutions beyond singularities

Proved no-loss-of-topology at finite time

Decomposed arbitrary maps into branched minimal immersions

Abstract

We analyse finite-time singularities of the Teichm\"uller harmonic map flow -- a natural gradient flow of the harmonic map energy -- and find a canonical way of flowing beyond them in order to construct global solutions in full generality. Moreover, we prove a no-loss-of-topology result at finite time, which completes the proof that this flow decomposes an arbitrary map into a collection of branched minimal immersions connected by curves.