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 ${\Sigma }^m\subset {\mathbb{R}}^n$ of class $C^1$ and of arbitrary dimension and codimension (or, more generally, an Ahlfors-regular compact set $\Sigma$ satisfying a mild general condition relating the size of holes in $\Sigma$ to the flatness of $\Sigma$ measured in terms of beta numbers) is in fact an embedded manifold of class $C^{1,\tau }\cap W^{2,p}$, where $p>m$ and $\tau =1-m/p$. 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 $\Sigma$ or (b) the size of spheres tangent to $\Sigma$ at one point and passing through another point of $\Sigma$. Appropriately defined \emph{maximal functions} of such integrands turn out to be of class $L^p(\Sigma )$ for $p>m$ if and only if the local graph representations of $\Sigma$ have second order derivatives in $L^p$ and $\Sigma$ 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 $\Sigma$ is a round sphere.

44 pages, 2 figures; several minor corrections

References in corpus (6)

• Tangent-point self-avoidance energies for curves
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• Sharp Boundedness and Regularizing effects of the integral Menger curvature for submanifolds
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Cited by in corpus (7)

• On some knot energies involving Menger curvature
• Geometric Sobolev-like embedding using high-dimensional Menger-like curvature
• Sharp Boundedness and Regularizing effects of the integral Menger curvature for submanifolds
• Compactness and isotopy finiteness for submanifolds with uniformly bounded geometric curvature energies
• Self-repulsiveness of energies for closed submanifolds
• M{ö}bius-invariant self-avoidance energies for non-smooth sets in arbitrary dimensions
• A Reifenberg type characterization for $m$-dimensional $C^1$-submanifolds of ${\mathbb{R}}^n$