Constant mean curvature invariant surfaces and extremals of curvature energies

摘要

We determine the extremal curves of curvature energy functionals which generalize a variational problem studied by Blaschke in R-3. 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. (C) 2018 Elsevier Inc. All rights reserved.