In this paper, we prove that the small energy harmonic maps from $\Bbb H^2$ to $\Bbb H^2$ are asymptotically stable under the wave map equation in the subcritical perturbation class. This result may be seen as an example supporting the soliton resolution conjecture for geometric wave equations without equivariant assumptions on the initial data. In this paper, we construct Tao's caloric gauge in the case when nontrivial harmonic map occurs. With the "dynamic separation" the master equation of the heat tension field appears as a semilinear magnetic wave equation. By the endpoint and weighted Strichartz estimates for magnetic wave equations obtained by the first author \cite{Lize1}, the asymptotic stability follows by a bootstrap argument.

中文翻译：

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波图方程下二维双曲空间间调和图的渐近稳定性。二、小能源案例

在本文中，我们证明了从$\Bbb H^2$到$\Bbb H^2$的小能量谐波映射在亚临界扰动类的波映射方程下是渐近稳定的。这个结果可以看作是支持几何波动方程的孤子分辨率猜想的一个例子，而没有对初始数据的等变假设。在本文中，我们构建了在非平凡调和映射出现的情况下的道的热量规。通过“动态分离”，热张力场的主方程表现为半线性磁波方程。通过第一作者 \cite{Lize1} 获得的磁波方程的端点和加权 Strichartz 估计，渐近稳定性遵循自举论证。