Harmonic Maps of Conic Surfaces With Cone Angles Less Than 2π

We prove the existence and uniqueness of harmonic maps in degree one homotopy classes of closed, orientable surfaces of positive genus, where the target has non-positive Gauss curvature and conic points with cone angles less than 2π. For a homeomorphism w of such a surface, we prove existence and uniqueness of minimizers in the homotopy class of w relative to the inverse images of the cone points with cone angles less than or equal to π. The latter can be thought of as minimizing maps from punctured Riemann surfaces into conic surfaces. We discuss the regularity of these maps near the inverse images of the cone points in detail. For relative minimizers, we relate the (non-zero) gradient of the energy functional with the Hopf differential. When the genus is zero, we prove the same relative minimization provided there are at least three cone points of cone angle less than or equal to π.