Remarks on approximate harmonic maps in dimension two

TL;DR

This paper investigates approximate harmonic maps from surfaces to manifolds, establishing energy identities, regularity results, and bubble tree convergence under specific boundedness conditions on tension fields.

Contribution

It proves energy identity and regularity results for approximate harmonic maps with tension fields in Morrey and L log L spaces, and establishes bubble tree convergence.

Findings

Energy identity holds for weakly convergent approximate harmonic maps.

Approximate harmonic maps with tension fields in L log L and Morrey spaces are in W^{2,1}.

Bubble tree convergence is established under bounded tension field conditions.

Abstract

For the class of approximate harmonic maps $u\in W^{1,2}(\Sigma,N)$ from a closed Riemmanian surface $(\Sigma,g)$ to a compact Riemannian manifold $(N, h)$, we show that (i) the so-called energy identity holds for weakly convergent approximate harmonic maps $\{u_n\}:\Sigma\to N$, with tension fields $\tau(u_n)$ bounded in the Morrey space $M^{1,\delta}(\Sigma)$ for some $0\le\delta<2$; and (ii) if an approximate harmonic map $u$ has tension field $\tau(u)\in L\log L(\Sigma)\cap M^{1,\delta}(\Sigma)$ for some $0\le\delta<2$, then $u\in W^{2,1}(\Sigma, N)$. Based on these estimates, we further establish the bubble tree convergence, referring to energy identity both $L^{2,1}$ of gradients and $L^1$-norm of hessians and the oscillation convergence, for a weakly convergent sequence of approximate harmonic maps $\{u_n\}$, with tension fields $\tau(u_n)$ uniformly bounded in $M^{1,\delta}(\Sigma)$ for some $0\le\delta<2$ and uniformly integrable in $L\log L(\Sigma)$.