Characterizing $W^{2,p}$~submanifolds by $p$-integrability of global curvatures

TL;DR

This paper establishes geometric conditions under which a compact manifold with finite curvature energies is smoothly embedded, linking curvature integrability to Sobolev regularity and flatness properties.

Contribution

It provides a characterization of $W^{2,p}$ submanifolds via $p$-integrability of global curvatures, connecting geometric curvature energies with Sobolev regularity and flatness analysis.

Findings

Characterization of $W^{2,p}$ submanifolds through curvature integrability.

Equivalence between maximal function integrability and second derivatives in $L^p$.

Lower bound for tangent sphere curvature energy attained only by round spheres.

Abstract

We give sufficient and necessary geometric conditions, guaranteeing that an immersed compact closed manifold $\Sigma^m\subset \R^n$ 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.