A quantitative approach to the regularity of a Riemannian surface

TL;DR

This paper introduces intrinsic and extrinsic definitions to quantify the regularity of Riemannian surfaces, demonstrating their equivalence up to a constant depending on the Hölder exponent.

Contribution

It proposes two new definitions for measuring the regularity of $C^{2,eta}$ surfaces and proves their equivalence, advancing the understanding of surface regularity in Riemannian geometry.

Findings

The intrinsic definition uses the Hölder norm of Gauss curvature.

The extrinsic definition relies on local smooth representations of the metric.

Both definitions are shown to be equivalent up to a constant.

Abstract

We introduce two definitions with the purpose of quantifying the concept of a $C^{2,\alpha}$ surface for $0 < \alpha < 1$. The intrinsic definition is given in terms of the $\alpha$-H\"{o}lder norm of the Gauss curvature function. The extrinsic one relies on the existence of a smooth local representation of the Riemannian metric. We show that these definitions are equivalent up to a constant depending on $\alpha$.