Circles minimize most knot energies

Abrams, Aaron; Cantarella, Jason; Fu, Joseph H. G.; Ghomi, Mohammad; Howard, Ralph

doi:10.1016/s0040-9383(02)00016-2

Cited by 64 publications

(75 citation statements)

References 11 publications

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“…The same claim was demonstrated more recently in different ways in [1,3]; see also a local proof in [2]. It is natural to consider the maximal value of p for which the inequality holds.…”

Section: If γ(S) Describes a Planar Curve Of Parametrized By Arclengtsupporting

confidence: 59%

On the critical exponent in an isoperimetric inequality for chords

2007

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The problem of maximizing the L p norms of chords connecting points on a closed curve separated by arclength u arises in electrostatic and quantum-mechanical problems. It is known that among all closed curves of fixed length, the unique maximizing shape is the circle for 1 ≤ p ≤ 2, but this is not the case for sufficiently large values of p. Here we determine the critical value p c (u) of p above which the circle is not a local maximizer finding, in particular, that p c ( If Γ(s) describes a planar curve of parametrized by arclength s and L is its total lengthdescribes the L p -mean of the Euclidean length of the chords connecting points separated by arclength u. A reasonable geometric question is to determine the shape that maximizes this quantity for any given value of p. Some phys-1

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“…The same claim was demonstrated more recently in different ways in [1,3]; see also a local proof in [2]. It is natural to consider the maximal value of p for which the inequality holds.…”

Section: If γ(S) Describes a Planar Curve Of Parametrized By Arclengtsupporting

confidence: 59%

On the critical exponent in an isoperimetric inequality for chords

2007

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“…So both, the energy and the length of the curve, is uniformly bounded in time. As Abrams et al [1] have shown that…”

Section: E α (C(t)) + λL(c(t)) ≤ E α (C(0)) + λL(c(0))mentioning

confidence: 91%

“…Abrams et al proved in [1] that for p ≥ 1 and j p − 1 < 2 p these energies are minimized by circles and that these energies are infinite for every closed regular curve if j p − 1 ≥ 2 p. 1 There is a reason why for the rest of our questions we will only consider the case p = 1: For p = 1, we expect that the first variation of E j, p leads to a degenerate elliptic operator of fractional order-even after breaking the symmetry of the equation coming from the invariance under re-parameterizations. We will only consider the nondegenerate case p = 1 and look at the one-parameter family We leave the case p = 1 for a later study.…”

Section: Introductionmentioning

confidence: 96%

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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“…Later on, Freedman, He, and Wang [1994] coined the name Möbius energy due to their seminal discovery that it is in fact invariant under Möbius transformations. The immediate consequence that circles are unique minimizers among all closed curves could be generalized by analytical means to the entire family (1) by Abrams, Cantarella, Fu, Ghomi, and Howard [2003].…”

Section: O'hara's Energiesmentioning

confidence: 99%

How nice are critical knots? Regularity theory for knot energies

Blatt

Reiter

2014

J. Phys.: Conf. Ser.

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Abstract. In this note we report on some recent developments in geometric knot theory which aims at finding links between geometric properties of a given knotted curve and its knot type. The central object of this field are so-called knot energies which are defined on closed embedded curves.First we present three important examples of two-parameter knot energy families, namely O'Hara's energies, the (generalized) integral Menger curvature, and the (generalized) tangentpoint energies.Subsequently we outline the main steps that lead to C ∞ -regularity of stationary pointsespecially minimizers-in the non-degenerate sub-critical range of parameters.Particular attention is devoted to the appearing parallels between these energies which, surprisingly at first glance, are quite similar from an analyst's perspective.

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Circles minimize most knot energies

Cited by 64 publications

References 11 publications

On the critical exponent in an isoperimetric inequality for chords

On the critical exponent in an isoperimetric inequality for chords

The gradient flow of O’Hara’s knot energies

How nice are critical knots? Regularity theory for knot energies

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