Harmonic nonholomorphic maps from a surface to a sphere

Abstract:

For every p and $\mathcal {D}$ such that $|\mathcal {D}| \leqslant p - 1$, there exist a Riemann surface of genus p and a harmonic nonholomorphic map of degree $\mathcal {D}$ from that surface to the sphere. For some of the degrees, a holomorphic map exists for the same Riemann surface.