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I'm no expert in this field, but I am familiar with the classification of rotationally symmetric surfaces with constant mean curvature by Delaunay. I am aware that once we drop embeddedness and smoothness, the situation becomes very complicated.

My question is, other than the family of surfaces from Delaunay, is there a classification of (embedded in a Riemannain manifold $M^n$, smooth) surfaces with constant mean curvature? If not, is there a classification known under additional conditions? $M^n = R^n$, closed surfaces, surfaces with boundary, finite total curvature, etc?

I write such a broad question as I find this topic quite interesting and beautiful, and hope that others also share this feeling.

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There's a remarkable correspondence between cmc surfaces in $R^3$ and minimal surfaces in $S^3$, due to Lawson. However, it only holds locally (or for simply-connected immersed surfaces), so usually doesn't give much information on embedded CMC surfaces in $R^3$, even though many embedded minimal surfaces in $S^3$ are known. – Ian Agol Dec 25 '10 at 0:08
This is interesting also, thank you. – Glen Wheeler Dec 25 '10 at 9:48
up vote 8 down vote accepted

Karsten Grosse-Brauckmann, Rob Kusner, and John Sullivan have written on the classification of embedded CMC surfaces for quite some time. I think a reasonable place to read about this program is this survey by Rob Kusner. You might also want to look at some of their other papers on the arxiv as well.

A nice idea used in their work which goes back to a paper of Korevaar, Kusner and Solomon in the 80's is that embdedded CMC surfaces have "ends" which asymptote to Delaunay surfaces - furthermore these ends can be assigned a "tension" (this should be familiar to you as the parameter which distinguishes between Delaunay surfaces) and these tensions satisfy a force balance rule.

I think the story for minimal surfaces is much better understood due to the work of Meeks and Minicozzi though I haven't read their work in much detail.

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the version of the survey on Kusner's webpage is slightly newer – j.c. Dec 25 '10 at 1:47
Thanks, this is very helpful. – Glen Wheeler Dec 25 '10 at 9:44

You might also look at the work of Katsuei Kenmotsu. His paper "Surfaces of revolution with prescribed mean curvature." (1980) generalizes the results of Delaunay to non-constant mean curvature functions. Meanwhile, the book "Surfaces With Constant Mean Curvature" investigates CMC surfaces in more general setting, and includes quite a few graphics and examples to titillate your senses :) (I agree with you that the subject is quite beautiful). Though I believe that most, if not all, of his investigations have been in the setting of $\mathbb{R}^3$...

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Let me add some uniqueness theorems for CMC and minimal surfaces:

1) A classical theorem of Hopf says that any immersed CMC sphere in $\mathbb{R}^3$ is the round sphere.

2) A classical theorem of Aleksandrov says that any embedded closed hypersurface in Euclidean space with constant mean curvature is the round sphere.

3) Very recently, Simon Brendle proved the Lawson conjecture: Any embedded minimal torus in $S^3$ is congruent to the Clifford torus, see arXiv:1203.6597v2.

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I would like to add a comment to Mr. Haslhofer's answer: This was conjectured by H.Hopf that any immersed surface of constant mean curvature is the round sphere, however, a counterexample by H.Wente proved it incorrect. – S.A.A Jun 22 '12 at 3:40

Since my previous answer to this question major progress was made: As Robert Haslhofer already mentioned, Brendle proved the Lawson conjecture that the only minimal CMC torus in the 3-sphere is the Clifford torus. Building on this work, Andrews and Li ("Embedded constant mean curvature tori in the three-sphere." J. Differential Geom. 99 (2015)) classified all embedded CMC tori in $S^3.$ To be more concrete, they used the same two point function as in Brendles proof in order to show that every embedded CMC torus must be rotationally symmetric, and hence is classified and can be written down in terms of elliptic functions explicitly.

The case of higher genus CMC surfaces in space forms is more difficult and unsolved by now. But there has been done computer experiments which suggest that the space of embedded CMC surfaces of higher genus $g\geq2$ with certain symmetries is related to the space of embedded CMC tori, see and the following image:

enter image description here

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Awesome, thanks for taking the time to add this! – Glen Wheeler Jan 25 at 5:46
Dear Glen, you are very welcome. – Sebastian Jan 25 at 18:21

Let me describe the case of CMC tori in $\mathbb{R}^3$ or $S^3:$ There does not exist a complete classification, but it was shown by Hitchin and Pinkall/Sterling independently that all these CMC tori are given (explicitly) in terms of algebro-geometric data defined on a compact Riemann surface, the so-called spectral curve. Associated to CMC surfaces is a holomorphic family of flat connections. These connections reduce due to the abelian nature of the fundamental group of a torus to the direct sum of flat line bundle connections. The spectral curve parametrizes these flat line bundles, and it can be shown that the spectral curve is compact. The log derivative of the holonomy (as a function) of the flat connections is given by an abelian differential with prescribed poles. Moreover, the eigenlines of the holonomy (with respect to a point on the torus) determine a holomorphic line bundle, which flows linearly in the Picard variety when the point on the torus flows.

The remaining problem is to determine the possible spectral curves, i.e. those who give rise to closed surfaces without periods. For CMC tori in $\mathbb{R}^3$ the possible spectral curve are somehow dense like the rationals in the reals, and for CMC tori in $S^3$ the possible spectral curves can be deformed smoothly. But a complete classification is still open.

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