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.



Is there a complete classification of constant mean curvature surfaces?
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](http://arxiv.org/abs/math/0207160). 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.


