Generalized neck analysis of harmonic maps from surfaces (Q2036272)

The sequence of harmonic maps from surfaces of uniformly bounded energy on the generalized neck domain is studied. The generalized neck domain is defined as a union of ghost bubbles and annular neck domains, which connects nontrivial bubbles. The main result of the present work is the following Theorem. Let $\Omega_i$ be a generalized neck domain, let $u_i$ be a sequence of harmonic maps satisfying the following two conditions: \begin{itemize} \item[(1)] the energy $\int_B|\nabla u_i|^2 dx$ is uniformly bounded (here $B$ is a unit ball in $R^2$); \item[(2)] for any $r>0$, $u_i$ converges smoothly to $u_\infty$ on $B\setminus B_r$, where $B_r$ is the ball of radius $r$ centered in the origin, and \[ \lim_{r\to o}\lim_{i\to\infty}\int_{B_r}|\nabla u_i|^2 dx>0. \] \end{itemize} Then there is a constant $C$ such that on $\Omega_i$, \[ |\nabla u_i|_{g_i}\leq C, \] where $g_i$ is a conformal metric defined on $\Omega_i$ in the terms of complex coordinate $z$ by the formula \[ g_i=\left( 1+\sum_{j=1}^i \frac{(\lambda_i^{(j)})^2}{|z-x_i^{(j)}|^4}\right)dz\wedge d\bar{z}. \]