Convergence of harmonic maps on the Poincaré disk

Abstract:

Let $\{f_n:{\mathbb D}\to { \mathbb D}\}$ be a sequence of locally quasiconformal harmonic maps on the unit disk ${\mathbb D}$ with respect to the Poincaré metric. Suppose that the energy densities of $f_n$ are uniformly bounded from below by a positive constant and locally uniformly bounded from above. Then there is a subsequence of $\{f_n\}$ that locally uniformly converges on ${\mathbb D}$, and the limit function is either a locally quasiconformal harmonic map of the Poincaré disk or a constant. Especially, if the limit function is not a constant, the subsequence can be chosen to satisfy some stronger conditions. As an application, it is proved that every point of the space $T_0({\mathbb D})$, a subspace of the universal Teichmüller space, can be represented by a quasiconformal harmonic map that is an asymptotic hyperbolic isometry.