Conformal immersions of prescribed mean curvature in R^3

TL;DR

This paper proves the existence of conformal immersions of spheres into three-dimensional space with a prescribed mean curvature, addressing affine indeterminacy and extending results to other space forms and higher genus surfaces.

Contribution

It establishes the existence of conformal immersions with prescribed mean curvature in R^3 and other space forms, including branched immersions and results for higher genus surfaces.

Findings

Existence of conformal immersions with prescribed mean curvature in R^3.

Extension of results to space forms S^3 and H^3.

Partial results for surfaces of higher genus.

Abstract

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.