Conformal immersions of prescribed mean curvature in R^3

Understand

We prove the existence of (branched) conformal immersions F: S^2 -> R^3 with mean curvature H > 0 arbitrarily prescribed up to a 3-dimensional affine indeterminacy.

• A similar result is proved for the space forms S^3, H^3 and partial results for surfaces of higher genus.