Open Access
September 2016 A frame energy for immersed tori and applications to regular homotopy classes
Andrea Mondino, Tristan Rivière
Author Affiliations +
J. Differential Geom. 104(1): 143-186 (September 2016). DOI: 10.4310/jdg/1473186541


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.


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Andrea Mondino. Tristan Rivière. "A frame energy for immersed tori and applications to regular homotopy classes." J. Differential Geom. 104 (1) 143 - 186, September 2016.


Published: September 2016
First available in Project Euclid: 6 September 2016

zbMATH: 1356.53008
MathSciNet: MR3544288
Digital Object Identifier: 10.4310/jdg/1473186541

Rights: Copyright © 2016 Lehigh University

Vol.104 • No. 1 • September 2016
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