# Immersed surface with circle as a boundary

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

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There are lots of immersed surfaces with boundary the circle. i think you are asking about immersed surfaces with constant mean curvature. As far as I know this is still an open question. –  Rbega Jan 9 '13 at 0:13
Yes, you're right. I didn't ask the question correctly but it fixes now. Do you know some references or progress about this question?. –  Mario Jan 9 '13 at 0:19
By the way thanks Rbega. –  Mario Jan 9 '13 at 0:20

Check out this paper of Rafael Lopez and references therein.

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Right, R. Lopez has several papers with progress towards the conjectures. See theorem 9.2 in the survey I linked to for a comprehensive list of all known conditions which guarantee an immersed surface with circular boundary to be a spherical cap. –  Gjergji Zaimi Jan 9 '13 at 1:05