Mobius Energy of Knots and Unknots

Freedman, Michael; He, Zheng-Xu; Wang, Zhenghan

doi:10.2307/2946626

Cited by 199 publications

(225 citation statements)

References 14 publications

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“…Like the Willmore energy, the Möbius energy has the remarkable property of being invariant under conformal transformations of R 3 [9]. In the case of knots other energies were considered by O'Hara [23].…”

Section: 2mentioning

confidence: 99%

“…It was conjectured by Freedman, He and Wang [9], in 1994, that the Möbius energy should be minimized, among the class of all nontrivial links in R 3 , by the stereographic projection of the standard Hopf link. The standard Hopf link (γ 1 ,γ 2 ) is described bŷ γ 1 (s) = (cos s, sin s, 0, 0) ∈ S 3 andγ 2 (t) = (0, 0, cos t, sin t) ∈ S 3 , and it is simple to check that E(γ 1 ,γ 2 ) = 2π 2 .…”

Section: 2mentioning

confidence: 99%

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Min-max theory, Willmore conjecture and the energy of links

Marques

Neves

2013

Bull Braz Math Soc, New Series

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Dedicated to the 60th anniversary of the founding of IMPA.Abstract. In this paper we give an overview of some aspects of the min-max theory of minimal surfaces, and discuss recent applications to conformally invariant problems in Geometry and Topology. The goal is to explain what the proofs of the Willmore conjecture for surfaces and the Freedman-He-Wang conjecture for links share in common. This is based on joint work of the authors [19] and on joint work of I. Agol and the authors [1].

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Section: 2mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Min-max theory, Willmore conjecture and the energy of links

Marques

Neves

2013

Bull Braz Math Soc, New Series

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“…In [14], Freedman, He, and Wang showed that even E 2 can be minimized within every prime knot class. Whether or not the same is true for composite knot classes is an open problem, though there are some numerical experiments that indicate that this might not be the case in every such knot class [19].…”

Section: Introductionmentioning

confidence: 99%

“…Whether or not the same is true for composite knot classes is an open problem, though there are some numerical experiments that indicate that this might not be the case in every such knot class [19]. Furthermore, they derived a formula for the L 2 -gradient of the Möbius energy [14,Equation 6.12] which was extended by Reiter [24,Theorem 1.45] to the energies E α for α ∈ [2,3). They showed that the first variation of these functionals can be given by Using the Möbius invariance of E 2 , Freedman, He, and Wang showed that local minimizers of the Möbius energy are of class C 1,1 [14]-and thus gave a first answer to the question about the niceness of the optimal shapes.…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, they derived a formula for the L 2 -gradient of the Möbius energy [14,Equation 6.12] which was extended by Reiter [24,Theorem 1.45] to the energies E α for α ∈ [2,3). They showed that the first variation of these functionals can be given by Using the Möbius invariance of E 2 , Freedman, He, and Wang showed that local minimizers of the Möbius energy are of class C 1,1 [14]-and thus gave a first answer to the question about the niceness of the optimal shapes. Zheng-Xu He combined this with a sophisticated bootstrapping argument to find that minimizers of the Möbius energy are of class C ∞ [17].…”

Section: Introductionmentioning

confidence: 99%

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The gradient flow of O’Hara’s knot energies

Blatt

2017

Math. Ann.

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Jun O'Hara invented a family of knot energies E j, p , j, p ∈ (0, ∞), O'Hara in Topology Hawaii (Honolulu, HI, 1990). World Science Publication, River Edge 1992. We study the negative gradient flow of the sum of one of the energies E α = E α,1 , α ∈ (2, 3), and a positive multiple of the length. Showing that the gradients of these knot energies can be written as the normal part of a quasilinear operator, we derive short time existence results for these flows. We then prove long time existence and convergence to critical points.

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The Euler-Lagrange equation and heat flow for the Möbius energy

2000

Comm. Pure Appl. Math.

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We prove the following results: 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. 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. 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.

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Mobius Energy of Knots and Unknots

Cited by 199 publications

References 14 publications

Min-max theory, Willmore conjecture and the energy of links

Min-max theory, Willmore conjecture and the energy of links

The gradient flow of O’Hara’s knot energies

The Euler-Lagrange equation and heat flow for the Möbius energy

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