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Tuesday, March 31, 2015

**Abstract:** (joint work with A. Lawrie and S.-J. Oh) We consider the equivairant wave maps problem from the hyperbolic plane into two model rotationally symmetric targets, namely the two sphere and the hyperbolic plane itself. Due to the non-Euclidean geometry of the domain, these problem exhibits markedly different phenomena compared to its Euclidean counterpart. For instance, there exist a one parameter family of finite energy stationary solutions (harmonic maps) even in the case where the target of the wave map is negatively curved (i.e. $H^2$). Moreover, when the target is the sphere the spectrum of the linearized operator about certain stationary solutions possesses a gap eigenvalue, i.e., a simple eigenvalue in the gap $(0, 1/4)$ between $0$ and the essential spectrum of the Laplacian on hyperbolic plane. The existence of such a gap eigenvalue raises interesting questions about the asymptotic dynamics of the solutions to the corresponding wave equations. If time permits, we will discuss similar phenomena for an equivariant Yang-Mills problem on hyperbolic space.