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

Abstract

We consider repulsive potential energies $\E_q(Σ)$, whose integrand measures tangent-point interactions, on a large class of non-smooth $m$-dimensional sets $\Sigma$ in ${\mathbb{R}}^n.$ Finiteness of the energy $\E_q(Σ)$ 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 Hölder continuous manner with the optimal Hölder 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 $\E_q(Σ)$.

47 pages, 1 figure

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