Convergence to harmonic maps for the Landau-Lifshitz flows on two dimensional hyperbolic spaces

In this paper, we prove that the solution of the Landau-Lifshitz flow $u(t,x)$ from $\mathbb{H}^2$ to $\mathbb{H}^2$ converges to some harmonic map as $t\to\infty$. The essential observation is that although there exist infinite numbers of harmonic maps from $\Bbb H^2$ to $\Bbb H^2$, the heat flow initiated from $u(t,x)$ for any given $t>0$ converges to the same harmonic map as the heat flow initiated from $u(0,x)$. This observation enables us to construct a variant of Tao's caloric gauge to reduce the convergence to harmonic maps for the Landau-Lifshitz flow to the decay of the corresponding heat tension field. The advantage of the strategy used in this paper is that we can see the limit harmonic map directly by evolving $u(0,x)$ along a heat flow without evolving the Landau-Lifshitz flow to the infinite time.