Asymptotics of the Teichm\"uller harmonic map flow

TL;DR

This paper analyzes the long-term behavior of the Teichmüller harmonic map flow, revealing that domain degeneration leads to a branched minimal immersion, through advanced geometric analysis of holomorphic quadratic differentials.

Contribution

It provides a detailed geometric analysis explaining the asymptotic behavior of the flow when the domain metric degenerates at infinite time.

Findings

Degeneration of the domain metric results in a branched minimal immersion.

Holomorphic quadratic differentials are key to understanding the flow's asymptotics.

The flow's long-term behavior is characterized by a specific geometric limit.

Abstract

The Teichm\"uller harmonic map flow, introduced in [9], evolves both a map from a closed Riemann surface to an arbitrary compact Riemannian manifold, and a constant curvature metric on the domain, in order to reduce its harmonic map energy as quickly as possible. In this paper, we develop the geometric analysis of holomorphic quadratic differentials in order to explain what happens in the case that the domain metric of the flow degenerates at infinite time. We obtain a branched minimal immersion from the degenerate domain.