Immersed surface with circle as a boundary - MathOverflow

Immersed surface with circle as a boundary

2013-01-08T23:25:22Z

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

Thanks Mario

Answer by Igor Rivin for Immersed surface with circle as a boundary

2013-01-09T00:57:15Z

Check out this paper of Rafael Lopez and references therein.

Answer by Gjergji Zaimi for Immersed surface with circle as a boundary

2013-01-09T01:00:31Z

Assuming your interest is in constant mean curvature surfaces with circular boundary, I found the survey, "Surfaces with constant mean curvature in Euclidean space" by R. Lopez to be a great introduction, and it contains the state of the art, and several references.

Hopf proved that the only constant mean curvature closed surfaces of genus 0 are spheres. Alexandrov's theorem says that constant mean curvature closed surfaces that are embedded are spheres. Unfortunately when we allow boundaries both analogs are conjectural:

Conjecture 1: The only constant mean curvature compact surfaces with circular boundary that are topological disks are spherical caps.

Conjecture 2: The only constant mean curvature compact surfaces with circular boundary that are embedded are spherical caps.