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

TL;DR

This paper introduces a new energy functional called Frame energy for immersed tori in Euclidean spaces, establishing lower bounds, analyzing critical points, and applying these results to classify immersions in regular homotopy classes.

Contribution

It defines the Frame energy linked to the Willmore energy, proves a lower bound similar to the Willmore conjecture, and applies the theory to classify immersed tori in regular homotopy classes.

Findings

Frame energy has a lower bound of 2π² for immersed tori in R^m.

Equality in the lower bound characterizes Clifford tori in R^4.

Critical points of the Frame energy are smooth due to hidden conservation laws.

Abstract

The paper is devoted to study the Dirichelet 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 first result, a Willmore-conjecture type lower bound is established : namely for every torus immersed in $\R^m$, $m\geq 3$, and any moving frame on it, the frame energy is at least $2\pi^2$ and equalty 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. Smootheness 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 $\R^3$ is performed.