Strict type-II blowup in harmonic map flow

TL;DR

This paper proves that at a strict type-II finite-time singularity in 2D harmonic map flow, the body map remains H"older continuous, providing insight into the singularity's structure.

Contribution

It establishes the H"older continuity of the body map at strict type-II blowups in 2D harmonic map flow, a new regularity result for such singularities.

Findings

The body map is H"older continuous at strict type-II blowup.

Defines strict type-II blowup based on energy scale behavior.

Provides a regularity property of the harmonic map flow near singularities.

Abstract

A finite-time singularity of 2D harmonic map flow will be called "strictly type-II" if the outer energy scale satisfies $\lambda(t) = O(T - t)^{\frac{1 + \alpha}{2}}.$ We prove that the body map at a strict type-II blowup is H\"older continuous.