Geodesic
On metrics arising on surfaces of constant mean curvature
Matematičeskie zametki, Tome 77 (2005) no. 4, pp. 617-622

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We formulate necessary and sufficient conditions on a Riemannian metric that ensure its embeddability in a three-dimensional space of constant curvature as a surface of constant mean curvature. This theorem is a generalization of a number of classical results, in particular, the Ricci theorem, which gives a description of metrics arising on minimal surfaces in $\mathbb R^3$. 
@article{MZM_2005_77_4_a13,
     author = {V. T. Fomenko},
     title = {On metrics arising on surfaces of constant mean curvature},
     journal = {Matemati\v{c}eskie zametki},
     pages = {617--622},
     publisher = {mathdoc},
     volume = {77},
     number = {4},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2005_77_4_a13/}
}
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V. T. Fomenko. On metrics arising on surfaces of constant mean curvature. Matematičeskie zametki, Tome 77 (2005) no. 4, pp. 617-622. http://geodesic.mathdoc.fr/item/MZM_2005_77_4_a13/