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The Euler‐Lagrange equation and heat flow for the Möbius energy
Author(s) -
He ZhengXu
Publication year - 2000
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/(sici)1097-0312(200004)53:4<399::aid-cpa1>3.0.co;2-d
Subject(s) - mathematics , integrable system , curvature , euclidean space , riemannian manifold , mathematical analysis , cube (algebra) , euclidean geometry , hyperbolic space , pure mathematics , geometry
We prove the following results: 1 A unique smooth solution exists for a short time for the heat equation associated with the Möbius energy of loops in a euclidean space, starting with any simple smooth loop. 2 A critical loop of the energy is smooth if it has cube‐integrable curvature. Combining this with an earlier result of M. Freedman, Z. Wang, and the author, we show that any local minimizer of the energy must be smooth. 3 Circles are the only two‐dimensional critical loops with cube‐integrable curvature.The technique also applies to a family of other knot energies. Similar problems are open for energies of surfaces or, more generally, for embedded submanifolds in a fixed Riemannian manifold. © 2000 John Wiley & Sons, Inc.