We show that Carathéodory's conjecture on umbilical points of closed convex surfaces may be reformulated in terms of the existence of at least one umbilical point in the graphs of functions f: R 2 → R ...

We extend the classical existence and uniqueness theory of Jenkins-Serrin (H = 0) and Spruck (H > 0) for the constant mean curvature equation over a domain in R², to domains in H² or S². This theory ...