Energy identity for the maps from a surface with tension field bounded in $L^p$

TL;DR

This paper proves an energy identity and neckless property for sequences of maps from a surface with bounded tension field in $L^p$, extending understanding of harmonic map blow-up behavior for certain target manifolds.

Contribution

It establishes the energy identity and neckless property for maps with tension fields in $L^p$, for $p geq 6/5$, a significant extension in harmonic map theory.

Findings

Energy identity during blow-up for $p geq 6/5$

Neckless property during blow-up

Applicable to general target manifolds

Abstract

Let $M$ be a closed Riemannian surface and $u_n$ a sequence of maps from $M$ to Riemannian manifold $N$ satisfying $$\sup_n(\|\nabla u_n\|_{L^2(M)}+\|\tau(u_n)\|_{L^p(M)})\leq \Lambda$$ for some $p>1$, where $\tau(u_n)$ is the tension field of the mapping $u_n$. For the general target manifold $N$, if $p\geq \frac 65$, we prove the energy identity and neckless during blowing up.