Willmore-Like Tori in Killing Submersions

The first variation formula and Euler-Lagrange equations for Willmore-like surfaces in Riemannian 3-spaces with potential are computed and, then, applied to the study of invariant Willmore-like tori with invariant potential in the total space of a Killing submersion. A connection with generalized elastica in the base surface of the Killing submersion is found, which is exploited to analyze Willmore tori in Killing submersions and to construct foliations of Killing submersions made up of Willmore tori with constant mean curvature.This research was partially supported by MINECO-FEDER Grants MTM2014-54804-P and MTM2013-47828-C2-1-P andGobiernoVasco Grant IT1094-16, Spain.The third author has been supported by Programa Predoctoral de Formaci´on de Personal Investigador No Doctor, GobiernoVasco, 2015