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

TL;DR

This paper introduces a high-dimensional Menger-like curvature and associated energies to analyze the geometric regularity of subsets in Euclidean space, establishing conditions under which these sets are smooth manifolds.

Contribution

It develops a new class of curvature energies based on Menger-like curvature, linking these energies to geometric regularity and smoothness of sets in Euclidean space.

Findings

Sets with finite curvature energy are Reifenberg flat.

Regularity results extend to C^{1,t} manifolds.

Energy controls local graph representations.

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 subset S of R^n we also introduce global versions of this discrete curvature, by taking supremum with respect to m+2-l points on S. 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 S. Next, we prove that if S is compact and m-Ahlfors regular and if p is greater than ml, then the P. Jones' \beta-numbers of S must decay as r^t with r \to 0 for some t in (0,1). If S 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,t} manifold. We also define a wide class of other sets for which this assertion is true. After that, we bootstrap the exponent t to the optimal one a = 1 - ml/p showing an analogue of the Morrey-Sobolev embedding theorem. Moreover, we obtain a qualitative control over the local graph representations of S only in terms of the energy.