Tangent-point repulsive potentials for a class of non-smooth $m$-dimensional sets in $\R^n$. Part I: Smoothing and self-avoidance effects

TL;DR

This paper introduces tangent-point repulsive potentials for non-smooth sets in Euclidean space, showing that finite energy enforces topological, geometric, and regularity properties, including the set being a $C^1$-manifold with Hölder continuous tangent planes.

Contribution

It establishes that finite tangent-point energy implies the set is a smooth $C^1$-manifold with Hölder continuous tangent planes, providing a new regularization and self-avoidance framework.

Findings

Finite energy excludes self-intersections.

Sets with finite energy have large projections and measure.

Finite energy sets are $C^1$-manifolds with Hölder continuous tangent planes.

Abstract

We consider repulsive potential energies $\E_q(\Sigma)$, whose integrand measures tangent-point interactions, on a large class of non-smooth $m$-dimensional sets $\Sigma$ in $\R^n.$ Finiteness of the energy $\E_q(\Sigma)$ has three sorts of effects for the set $\Sigma$: topological effects excluding all kinds of (a priori admissible) self-intersections, geometric and measure-theoretic effects, providing large projections of $\Sigma$ onto suitable $m$-planes and therefore large $m$-dimensional Hausdorff measure of $\Sigma$ within small balls up to a uniformly controlled scale, and finally, regularizing effects culminating in a geometric variant of the Morrey-Sobolev embedding theorem: Any admissible set $\Sigma$ with finite $\E_q$-energy, for any exponent $q>2m$, is, in fact, a $C^1$-manifold whose tangent planes vary in a H\"older continuous manner with the optimal H\"older exponent $\mu=1-(2m)/q$. Moreover, the patch size of the local $C^{1,\mu}$-graph representations is uniformly controlled from below only in terms of the energy value $\E_q(\Sigma)$.