Minimal Hölder regularity implying finiteness of integral Menger curvature

Kolasiński, Sławomir; Szumańska, Marta

doi:10.1007/s00229-012-0565-y

Cited by 12 publications

(13 citation statements)

References 20 publications

(24 reference statements)

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“…The regularizing behaviour of this energy extends to general measurable sets X ⊂ R n (instead of a curve γ), see Scholtes [2012] and references therein. For higher-dimensional objects one first has to find an appropriate notion for R. We refer to [Lerman & Whitehouse, 2011] (for M 2 ), [Strzelecki & von der Mosel, 2005 and [Blatt & Kolasiński, 2011;Kolasiński, 2012a,b;Kolasiński et al, 2012;Kolasiński & Szumańska, 2011].…”

Section: Integral Menger Curvaturementioning

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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Section: Integral Menger Curvaturementioning

confidence: 99%

How nice are critical knots? Regularity theory for knot energies

Blatt

Reiter

2014

J. Phys.: Conf. Ser.

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“…Knowing that Σ is a compact, closed, C 1,λ/κ -submanifold of R n , we prove that also the constant M θβ from the (θ β) condition can be replaced by an absolute constant. Then we obtain estimates on the oscillation of tangent planes of Σ solely in terms of E, m, l and p. This allows to prove that the size of a single patch of Σ representable as a graph of some function is controlled solely in terms of E, m, l and p. Next we bootstrap the exponent λ κ to the optimal one α = 1 − ml p (see [14] and [1] for the proof that this is indeed optimal). Theorem 3.…”

Section: Introductionmentioning

confidence: 99%

“…This work already lead to a few other results. In our joint work with Szumańska [14] we have constructed an example of a function f ∈ C 1,α 0 ([0, 1] m ), where α 0 = 1 − m(m+1) p , whose graph has infinite E m+2 p -energy and we proved that for any α 1 > α 0 the graphs of C 1,α 1 functions always have finite energy. Later this result was complemented by our joint work with Blatt [1], where we have shown that a C 1 -submanifold of R n has finite E l p -energy for some p > m(l − 1) and l ∈ {2, .…”

Section: Introductionmentioning

confidence: 99%

Geometric Sobolev-like embedding using high-dimensional Menger-like curvature

Kolasiński

2014

Trans. Amer. Math. Soc.

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Abstract. We study a modified version of Lerman-Whitehouse Menger-like curvature defined for (m + 2) points in an n-dimensional Euclidean space. For 1 ≤ l ≤ m + 2 and an m-dimensional set Σ ⊂ R n we also introduce global versions of this discrete curvature, by taking supremum with respect to (m+2−l) points on Σ. We then define geometric curvature energies by integrating one of the global Menger-like curvatures, raised to a certain power p, over all l-tuples of points on Σ. Next, we prove that if Σ is compact and m-Ahlfors regular and if p is greater than the dimension of the set of all l-tuples of points on Σ (i.e. p > ml), then the P. Jones' β-numbers of Σ must decay as r τ with r → 0 for some τ ∈ (0, 1). If Σ is an immersed C 1 manifold or a bilipschitz image of such set then it follows that it is Reifenberg flat with vanishing constant, hence (by a theorem of David, Kenig and Toro) an embedded C 1,τ manifold. We also define a wide class of other sets for which this assertion is true. After that, we bootstrap the exponent τ to the optimal one α = 1 − ml/p showing an analogue of the Morrey-Sobolev embedding theorem W 2,p ⊆ C 1,α . Moreover, we obtain a qualitative control over the local graph representations of Σ only in terms of the energy.

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“…Menger curvature for higher-dimensional objects has been discussed in [19,21,22,4,20,34]. Further information on the context of the integral Menger curvature within the field of geometric knot theory and geometric curvature energies can be found in the recent surveys by Strzelecki and von der Mosel [38,37].…”

Section: Introductionmentioning

confidence: 99%

Towards a regularity theory for integral Menger curvature

Blatt¹,

Reiter²

2015

Ann. Acad. Sci. Fenn. Math.

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Abstract. We generalize the notion of integral Menger curvature introduced by Gonzalez and Maddocks [14] by decoupling the powers in the integrand. This leads to a new two-parameter family of knot energies intM (p,q) . We classify finite-energy curves in terms of Sobolev-Slobodeckiȋ spaces. Moreover, restricting to the range of parameters leading to a sub-critical Euler-Lagrange equation, we prove existence of minimizers within any knot class via a uniform bi-Lipschitz bound. Consequemtly, intM (p,q) is a knot energy in the sense of O'Hara. Restricting to the non-degenerate sub-critical case, a suitable decomposition of the first variation allows to establish a bootstrapping argument that leads to C ∞ -smoothness of critical points.

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Minimal Hölder regularity implying finiteness of integral Menger curvature

Cited by 12 publications

References 20 publications

How nice are critical knots? Regularity theory for knot energies

How nice are critical knots? Regularity theory for knot energies

Geometric Sobolev-like embedding using high-dimensional Menger-like curvature

Towards a regularity theory for integral Menger curvature

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