A Reifenberg type characterization for $m$-dimensional $C^1$-submanifolds of $\mathbb R^n$

TL;DR

This paper offers a Reifenberg type characterization for smooth submanifolds in Euclidean space, linking flatness conditions with convergence of approximating planes and providing integral criteria for such structures.

Contribution

It introduces a new Reifenberg type characterization for $C^1$-submanifolds, connecting flatness, convergence, and integral conditions, expanding understanding of geometric regularity.

Findings

Characterization equivalent to Reifenberg-flatness with vanishing constant

Integral condition involving $ heta(r)$-numbers as sufficient criterion

Examples showing the integral condition is not necessary

Abstract

We provide a Reifenberg type characterization for $m$-dimensional $C^1$-submanifolds of $\mathbb R^n$. This characterization is also equivalent to Reifenberg-flatness with vanishing constant combined with suitably converging approximating $m$-planes. Moreover, a sufficient condition can be given by the finiteness of the integral of the quotient of $\theta(r)$-numbers and the scale $r$, and examples are presented to show that this last condition is not necessary.