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

Abstract

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.

How nice are critical knots? Regularity theory for knot energies

“…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].…”

“…The above-mentioned result can be refined as follows, providing an explicit characterization of finite-energy curves: the image of Γ is an embedded manifold of class W 2−1/q,q ⊂ C 1,1−2/q if and only if E q (Γ) is finite [Blatt, 2011]. Results for higher-dimensional analoga to E q can be found in [Strzelecki & von der Mosel, 2011b;Blatt, 2011;Kolasiński et al, 2012].…”

Towards a regularity theory for integral Menger curvature

“…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].…”

Regularity theory for tangent-point energies: The non-degenerate sub-critical case

“…The energy spaces are discussed in [7]. Energies for higher-dimensional objects are considered in Strzelecki and von der Mosel [52,53,56], Kolasiński [31,32], and Kolasiński, Strzelecki, and von der Mosel [33].…”