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

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.