Integral Menger curvature for sets of arbitrary dimension and codimension

Kolasiński, Sławomir

doi:10.48550/arxiv.1011.2008

Cited by 4 publications

(23 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We will not use this fact in this article. The proof for the E m+2 p -energy can be found in [13,Theorem 2.13].…”

Section: 1mentioning

confidence: 99%

“…Their idea was to use the topological linking number to prevent holes in Σ. Any admissible set in the sense of [26] with finite E l p -energy for some p > ml, satisfies the (θ β) condition (see [13,Theorem 4.15] for the case l = m + 2), hence, by Theorem 1, it is a closed C 1,λ/κ -manifold.…”

Section: Introductionmentioning

confidence: 99%

“…For any two spaces U and V in G (δ, H) there exists a continuous path γ : [0, 1] → G (δ, H) such that γ(0) = V and γ(1) = U . Corollary 3.14 (cf [13]. Corollary 4.3).…”

mentioning

confidence: 95%

See 2 more Smart Citations

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

Kolasiński

2014

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…We will not use this fact in this article. The proof for the E m+2 p -energy can be found in [13,Theorem 2.13].…”

Section: 1mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Kolasiński

2014

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…We can use the above formula to compare Menger curvature and its higher dimensional generalizations 1 . In this paper we use integral curvature functional E p defined in [4] whose integrand is the p-th power of the discrete curvature K. Definition 2.2. Let T = (x 0 , .…”

Section: Preliminariesmentioning

confidence: 99%

“…We also show that α0 is optimal by constructing examples of C 1,α 0 functions with graphs of infinite integral curvature. p (see [11] for the M p case and [4]…”

mentioning

confidence: 99%

Minimal Hölder regularity implying finiteness of integral Menger curvature

Kolasiński

Szumańska

2012

manuscripta math.

Self Cite

View full text Add to dashboard Cite

show abstract

Characterizing W 2,p Submanifolds by p -Integrability of Global Curvatures

2013

Self Cite

View full text Add to dashboard Cite

We give sufficient and necessary geometric conditions, guaranteeing that an immersed compact closed manifold Σ m ⊂ R n of class C 1 and of arbitrary dimension and codimension (or, more generally, an Ahlfors-regular compact set Σ satisfying a mild general condition relating the size of holes in Σ to the flatness of Σ measured in terms of beta numbers) is in fact an embedded manifold of class C 1,τ ∩ W 2,p , where p > m and τ = 1 − m/p. The results are based on a careful analysis of Morrey estimates for integral curvature-like energies, with integrands expressed geometrically, in terms of functions that are designed to measure either (a) the shape of simplices with vertices on Σ or (b) the size of spheres tangent to Σ at one point and passing through another point of Σ.Appropriately defined maximal functions of such integrands turn out to be of class L p (Σ) for p > m if and only if the local graph representations of Σ have second order derivatives in L p and Σ is embedded. There are two ingredients behind this result. One of them is an equivalent definition of Sobolev spaces, widely used nowadays in analysis on metric spaces. The second one is a careful analysis of local Reifenberg flatness (and of the decay of functions measuring that flatness) for sets with finite curvature energies. In addition, for the geometric curvature energy involving tangent spheres we provide a nontrivial lower bound that is attained if and only if the admissible set Σ is a round sphere.

show abstract

Integral Menger curvature for sets of arbitrary dimension and codimension

Cited by 4 publications

References 18 publications

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

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

Minimal Hölder regularity implying finiteness of integral Menger curvature

Characterizing W 2,p Submanifolds by p -Integrability of Global Curvatures

Contact Info

Product

Resources

About