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

摘要

We consider repulsive potential energies E_q(Σ), whose integrand measures tangent-point interactions, on a large class of non-smooth m-dimensional sets Σ in □~n. Finiteness of the energy E_q(Σ) has three sorts of effects for the set Σ: topological effects excluding all kinds of (a priori admissible) self-intersections, geometric and measure-theoretic effects, providing large projections of Σ onto suitable m-planes and therefore large m-dimensional Hausdorff measure of Σ 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 Σ with finite E_q-energy, for any exponent q>2m, is, in fact, a C ~1-manifold whose tangent planes vary in a H?lder continuous manner with the optimal H?lder exponent μ=1-(2m)/q. Moreover, the patch size of the local C ~(1,μ) -graph representations is uniformly controlled from below only in terms of the energy value E_q(Σ).