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We study time and space equivariant wave maps from $M\times\RR\rightarrowS^2,$ where $M$ is diffeomorphic to a two dimensional sphere and admits anaction of SO(2) by isometries. We assume that metric on $M$ can be written as$dr^2+f^2(r)d\theta^2$ away from the two fixed points of the action, where thecurvature is positive, and prove that stationary (time equivariant)rotationally symmetric (of any rotation number) smooth wave maps exist and arestable in the energy topology. The main new ingredient in the construction,compared with the case where $M$ is isometric to the standard sphere(considered by Shatah and Tahvildar-Zadeh \cite{ST1}), is the the use oftriangle comparison theorems to obtain pointwise bounds on the fundamentalsolution on a curved background.

Stability of Stationary Wave Maps from a Curved Background to a Sphere