Conformal immersions of prescribed mean curvature in R3R3

We prove the existence of (branched) conformal immersions F:S2→R3F:S2→R3 with mean curvature H>0H>0 arbitrarily prescribed up to a 3-dimensional affine indeterminacy. A similar result is proved for the space forms S3S3, H3H3 and partial results for surfaces of higher genus.