Sharp Boundedness and Regularizing effects of the integral Menger curvature for submanifolds

TL;DR

This paper characterizes embedded compact $C^1$ manifolds with finite integral Menger curvature as locally Sobolev-Slobodeckij graphs and extends these results to intermediate energies, revealing regularizing effects.

Contribution

It provides a new characterization of manifolds with finite integral Menger curvature using Sobolev-Slobodeckij spaces and explores properties of intermediate energies.

Findings

Finite integral Menger curvature characterizes certain $C^1$ manifolds.

Manifolds with finite energy are locally graphs of Sobolev-Slobodeckij functions.

Intermediate energies exhibit similar regularizing properties.

Abstract

In this paper we show that embedded and compact $C^1$ manifolds have finite integral Menger curvature if and only if they are locally graphs of certain Sobolev-Slobodeckij spaces. Furthermore, we prove that for some intermediate energies of integral Menger type a similar characterization of objects with finite energy can be given.