A frame energy for immersed tori and applications to regular homotopy classes

Abstract

The paper is devoted to studying the Dirichlet energy of moving frames on $2$-dimensional tori immersed in the Euclidean $3 \leq m$-dimensional space. This functional, called frame energy, is naturally linked to the Willmore energy of the immersion and on the conformal structure of the abstract underlying surface. As the first result, a Willmore-conjecture type lower bound is established: namely for every torus immersed in $\mathbb{R}^m, m \geq 3$, and any moving frame on it, the frame energy is at least $2\pi^2$ and equality holds if and only if $m \geq 4$, the immersion is the standard Clifford torus (up to rotations and dilations), and the frame is the flat one. Smoothness of the critical points of the frame energy is proved after the discovery of hidden conservation laws and, as application, the minimization of the frame energy in regular homotopy classes of immersed tori in $\mathbb{R}^3$ is performed.