Is there a solution or progress of the following problem (maybe old conjecture): An immersed surface with a circle as a boundary is a part of a sphere??. If we replace "immersed" by "embedded" I think the problem is solve by Alexandroff kind of long time ago (maybe we have also to assume that the mean curvature is constant). Is someone could enlighten about what exactly Alexandroff solves and what is to do, it would help me a lot.

Thanks Mario