  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.
