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 Sullivan 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 forec b 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.

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.