Refined asymptotics of the Teichm\"uller harmonic map flow into general targets

TL;DR

This paper studies the long-term behavior of the Teichmüller harmonic map flow, showing convergence to branched minimal immersions and analyzing the geometric structure of the flow's limits.

Contribution

It develops a compactness theory for the flow, proves no energy loss during convergence, and constructs an example with disconnected limit images.

Findings

Flow converges to branched minimal immersions at certain times.

Necks connecting images become arbitrarily thin over time.

No loss of energy in the convergence process.

Abstract

The Teichm\"uller harmonic map flow is a gradient flow for the harmonic map energy of maps from a closed surface to a general closed Riemannian target manifold of any dimension, where both the map and the domain metric are allowed to evolve. Given a weak solution of the flow that exists for all time $t\geq 0$, we find a sequence of times $t_i\to\infty$ at which the flow at different scales converges to a collection of branched minimal immersions with no loss of energy. We do this by developing a compactness theory, establishing no loss of energy, for sequences of almost-minimal maps. Moreover, we construct an example of a smooth flow for which the image of the limit branched minimal immersions is disconnected. In general, we show that the necks connecting the images of the branched minimal immersions become arbitrarily thin as $i\to\infty$.