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A frame energy for immersed tori and applications to regular homotopy classes
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Mondino, Andrea and Rivière, Tristan (2016) A frame energy for immersed tori and applications to regular homotopy classes. Journal of Differential Geometry, 104 (1). pp. 143-186. doi:10.4310/jdg/1473186541 ISSN 0022-040X.
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Official URL: http://dx.doi.org/10.4310/jdg/1473186541
Abstract
The paper is devoted to studying the Dirichlet energy of moving frames on 2-dimensional tori immersed in the Euclidean 3≤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 Rm,m≥3, and any moving frame on it, the frame energy is at least 2π2 and equality holds if and only if m≥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 R3 is performed.
| Item Type: | Journal Article | ||||||
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| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||
| Journal or Publication Title: | Journal of Differential Geometry | ||||||
| Publisher: | Lehigh University * Department of Mathematics | ||||||
| ISSN: | 0022-040X | ||||||
| Official Date: | 1 September 2016 | ||||||
| Dates: |
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| Volume: | 104 | ||||||
| Number: | 1 | ||||||
| Page Range: | pp. 143-186 | ||||||
| DOI: | 10.4310/jdg/1473186541 | ||||||
| Status: | Peer Reviewed | ||||||
| Publication Status: | Published | ||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||
| Date of first compliant deposit: | 20 June 2019 | ||||||
| Date of first compliant Open Access: | 21 June 2019 | ||||||
| RIOXX Funder/Project Grant: |
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