Harmonic maps into real hyperbolic space

Abstract

In [2,4,5,6,7 ] Calabi, Borbosa and Chern showed that there is a one-to-one correspondence between arbitrary pairs of full isotropic (terminology as in [8]) harmonic maps $\pm\phi:M→ S^2m$ from a Riemann surface to Euclidean sphere and full totally isotropic holomorphic maps $f:M→ℂP^2m$ from the surface to complex projective space.In this paper we show, very explicity, how to construct a similar one-to-one correspondence when $S^2m$ is replaced by real hyperbolic space $H^2m$ with its standard metric. We get over a difficulty encountered by Barbosa of dealing with the zeros of certain wedge product by a technique adapted from [8].(The case of indefinite complex hyperbolic and projective spaces will be considered in a separate paper).