Harmonic maps into conic surfaces with cone angles less than $2\pi$

We prove the existence and uniqueness of harmonic maps in degree one homotopy classes of closed, orientable surfaces of positive genus, when the target has conic points with cone angles less than $2\pi$. For a cone point $p$ of cone angle less than or equal $\pi$ we show that one can minimize, uniquely, in the relative homotopy class of a homeomorphism sending a fixed point $q$ in the domain to $p$. The latter can be interpreted as minimizing maps from punctured Riemann surfaces.