*Accepted Paper*

**Inserted:** 26 jul 2013

**Last Updated:** 24 mar 2016

**Journal:** Journal of Differential Geometry

**Year:** 2013

**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 ${\mathbb R}^m$, $m\geq 3$, 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 (up to a constant tangential rotation). 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 ${\mathbb R}^3$ is performed.

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