Immersed surface with circle as a boundary - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T21:27:29Z http://mathoverflow.net/feeds/question/118405 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/118405/immersed-surface-with-circle-as-a-boundary Immersed surface with circle as a boundary Mario 2013-01-08T23:25:22Z 2013-01-09T01:00:31Z <p>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.</p> <p>Thanks Mario</p> http://mathoverflow.net/questions/118405/immersed-surface-with-circle-as-a-boundary/118412#118412 Answer by Igor Rivin for Immersed surface with circle as a boundary Igor Rivin 2013-01-09T00:57:15Z 2013-01-09T00:57:15Z <p>Check out <a href="http://www.ugr.es/~rcamino/publications/pdf/art14.pdf" rel="nofollow">this paper of Rafael Lopez</a> and references therein.</p> http://mathoverflow.net/questions/118405/immersed-surface-with-circle-as-a-boundary/118413#118413 Answer by Gjergji Zaimi for Immersed surface with circle as a boundary Gjergji Zaimi 2013-01-09T01:00:31Z 2013-01-09T01:00:31Z <p>Assuming your interest is in constant mean curvature surfaces with circular boundary, I found the survey, <a href="http://www.ugr.es/~rcamino/publications/pdf/art57.pdf" rel="nofollow">"Surfaces with constant mean curvature in Euclidean space"</a> by R. Lopez to be a great introduction, and it contains the state of the art, and several references. </p> <p>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:</p> <blockquote> <p><strong>Conjecture 1:</strong> The only constant mean curvature compact surfaces with circular boundary that are topological disks are spherical caps.</p> <p><strong>Conjecture 2:</strong> The only constant mean curvature compact surfaces with circular boundary that are embedded are spherical caps.</p> </blockquote>