Weighted Lojasiewicz inequalities and regularity of harmonic map flow

TL;DR

This work analyzes finite-time singularities of the harmonic map flow in the critical dimension $2$ by linking a Gaussian-weighted Łojasiewicz inequality to regularity properties of the flow. The authors develop a weighted energy framework with $\Phi(u)=\tfrac12\int |du|^2 e^{-r^2/4}$ and a twisted stress-energy $\widehat{\mathcal{T}}$, proving a quantitative inequality $|\Phi(u)-4\pi n|\le C_{k,\beta}\|\widehat{\mathcal{T}}(u)\|_1^{2-\beta}$ when $E(u)\le 4\pi k$, together with a weighted Poincaré inequality that enables localization. They derive two Struwe-type monotonicity formulas in $2$D with Gaussian weights, producing differential inequalities that govern energy concentration and enabling control of energy density and oscillation. Using these tools, they obtain oscillation bounds near singular times and show that the body map $u(T)$ extends continuously and that bubble-tree decompositions are neckless, at least for maps into $S^2$, with precise Hölder or log-polynomial decay depending on the Łojasiewicz exponent. The results offer a robust regularity criterion for harmonic map flow near singularities and a pathway toward broader applicability to geometric evolution equations.

Abstract

At a finite-time singularity of harmonic map flow in the critical dimension, we show that a Lojasiewicz inequality between the quantities appearing in Struwe's monotonicity formula implies continuity of the body map and the no-neck property for bubble-tree decompositions. We prove such an inequality when the target is $S^2,$ yielding both properties in this case.

Theorems & Definitions (16)

• Lemma 2.1
• proof
• Theorem 2.2
• proof
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Lemma 4.1
• proof