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In this talk, we consider equivariant wave maps from the hyperbolic plane into two model rotationally symmetric targets, namely the two sphere ($\mathbb{S}^{2}$) and the hyperbolic plane itself ($\mathbb{H}^{2}$). Due to the non-Euclidean geometry of the domain, this problem exhibits markedly different phenomena compared to its Euclidean counterpart. For instance, there exist numerous stationary solutions to not only $\mathbb{S}^{2}$ but also $\mathbb{H}^{2}$, which has a negative constant curvature. Moreover, when the target is $\mathbb{S}^{2}$, the spectrum of the linearized operator about certain stationary solutions possesses a \emph{gap eigenvalue}, i.e., a simple eigenvalue in the gap $(0, 1/4)$ between $0$ and the essential spectrum. (Joint with A. Lawrie and S. Shahshahani.)