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May/June 2019 $L^2$ representations of the Second Variation and Łojasiewicz-Simon gradient estimates for a decomposition of the Möbius energy
Katsunori Gunji
Adv. Differential Equations 24(5/6): 321-376 (May/June 2019). DOI: 10.57262/ade/1554256827

Abstract

Knot energies, one of which is the Möbius energy, are constructed to measure the well-proportioned of the knot. The best-proportioned knot in the given knot class may be determined by the gradient flow of the energy. Indeed, Blatt showed the global existence and convergence of the gradient flow of the Möbius energy near stationary points. The Łojasiewicz inequality played an important role in proving this result. The inequality derived from $L^2$ representation of the first and second variations. On the other hand, Ishizeki and Nagasawa showed that the Möbius energy can be decomposed into three parts that are Möbius invariant. Each part has an $L^2$ representation of the first variation. In this paper, we discuss the $L^2$ representation of the second variation for each decomposed part of the Möbius energy, and derive explicit formulas for it. As a consequence of this and Chill's findings the Łojasiewicz inequality is derived.

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Katsunori Gunji. "$L^2$ representations of the Second Variation and Łojasiewicz-Simon gradient estimates for a decomposition of the Möbius energy." Adv. Differential Equations 24 (5/6) 321 - 376, May/June 2019. https://doi.org/10.57262/ade/1554256827

Information

Published: May/June 2019
First available in Project Euclid: 3 April 2019

zbMATH: 07197890
MathSciNet: MR3936013
Digital Object Identifier: 10.57262/ade/1554256827

Subjects:
Primary: 26D10 , 49J50 , 49Q10 , 53A04

Rights: Copyright © 2019 Khayyam Publishing, Inc.

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Vol.24 • No. 5/6 • May/June 2019
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