Constant mean curvature invariant surfaces and extremals of curvature energies

We determine the extremal curves of curvature energy functionals which generalize a variational problem studied by Blaschke in R3. The generalization is made by extending both, the lagrangian energy itself, and the ambient space (to riemannian and lorentzian 3-space forms). Then, we show that constant mean curvature (CMC) invariant surfaces in 3-space forms can be constructed, locally, as the evolution of the above extremals under their Binormal flow with appropriate velocity. Moreover, we see that these surfaces are intrinsically described by a warping function satisfying an Ermakov-Milne-Pinney equation. Finally, we use the previous findings to extend known results, on isometric deformations of CMC surfaces and the Lawson's correspondence of CMC cousins, to our background spaces.