Finite-time degeneration for variants of Teichm\"uller harmonic map flow

TL;DR

This paper investigates conditions under which solutions of variants of Teichmüller harmonic map flow from surfaces to targets can degenerate in finite time, establishing precise thresholds related to the stretching rate of the image of thin collars.

Contribution

It provides new criteria for finite-time degeneration of Teichmüller harmonic map flow, including sharp thresholds for the rescaled flow based on the stretching rate of the image.

Findings

Finite-time degeneration occurs when the image stretches out at a rate of at least inj(M,g)^{-(1/4 + δ)}.

For the rescaled flow, degeneration is inevitable if stretching exceeds |log(inj(M,g))|^{1/2 + δ}.

Finite-time degeneration can be avoided if the image stretches out no faster than |log(inj(M,g))|^{1/2}.

Abstract

We consider the question of whether solutions of variants of Teichm\"uller harmonic map flow from surfaces $M$ to general targets can degenerate in finite time. For the original flow from closed surfaces of genus at least $2$, as well as the flow from cylinders, we prove that such a finite-time degeneration must occur in situations where the image of thin collars is `stretching out' at a rate of at least $\text{inj}(M,g)^{-(\frac14+\delta)}$, and we construct targets in which the flow from cylinders must indeed degenerate in finite time. For the rescaled Teichm\"uller harmonic map flow, the condition that the image stretches out is not only sufficient but also necessary and we prove the following sharp result: Solutions of the rescaled flow cannot degenerate in finite time if the image stretches out at a rate of no more than $\lvert\log(\text{inj}(M,g))\rvert^{\frac12}$, but must degenerate in finite time if it stretches out at a rate of at least $\lvert\log(\text{inj}(M,g))\rvert^{\frac12+\delta}$ for some $\delta>0$.