Energy identity for approximate harmonic maps from surface to general targets

TL;DR

This paper establishes an energy identity and neckless property for approximate harmonic maps from surfaces to general targets under bounded energy and tension field conditions, highlighting the sharpness of these results with counterexamples.

Contribution

It proves the energy identity and neckless property for approximate harmonic maps with bounded energy and tension fields in L^p, extending previous results to more general targets.

Findings

Energy identity holds under specified conditions.

Neckless property is established during blow-up.

Results are sharp, as shown by counterexamples.

Abstract

Let $u_n$ be a sequence of mappings from a closed Riemannian surface $M$ to a general Riemannian manifold $N$. If $u_n$ satisfies \beno \sup_{n}\big(\|\nabla u_n\|_{L^2(M)}+\|\tau(u_n)\|_{L^{p}(M)}\big)\leq \Lambda\quad \text{for some}\,\,p>1, \eeno where $\tau(u_n)$ is the tension field of $u_n$, then there hold the so called energy identity and neckless property during blowing up. This result is sharp by Parker's example, where the tension fields of the mappings from Riemannian surface are bounded in $L^1(M)$ but the energy identity fails.