A quantitative approach to the regularity of a Riemannian surface

Matan Eilat

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arXiv · 2507.21921

A quantitative approach to the regularity of a Riemannian surface

Matan Eilat

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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 $\alpha$.

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Matan Eilat. 2025-07-29. A quantitative approach to the regularity of a Riemannian surface. https://arxiv.org/abs/2507.21921

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