Constant mean curvature surfaces are given by solutions of a variational problem and have been studied for long time.
On the one hand, it is known that the structure equation of the constant mean curvature surface is an integrable equation.
I am working on these constant mean curvature surfaces and their generalization, harmonic maps, using theory of integrable systems.

Papers

[1] S.-P. Kobayashi, Real forms of complex surfaces of constant mean curvature, Trans. Amer. Math. Soc. 363 (2011), no. 4, 1765–1788.

[2] J. F. Dorfmeister, J. Inoguchi, S.-P. Kobayashi, Constant mean curvature surfaces in hyperbolic 3-space via loop groups, J. Reine Angew. Math. 686 (2014), 1-36

[3] J. F. Dorfmeister, J. Inoguchi, S.-P. Kobayashi, A loop group method for affine harmonic maps into Lie groups, Advances in Mathematics, 298 (2016), 207-253.