On the existence of closed $C^{1,1}$ curves of constant curvature

TL;DR

This paper proves the existence of closed $C^{1,1}$ curves with constant curvature on any Riemannian surface, highlighting the regularity limitations of such curves through examples.

Contribution

It establishes the existence of immersed $C^{1,1}$ curves of constant curvature on all Riemannian surfaces and demonstrates the optimal regularity achievable.

Findings

Existence of $C^{1,1}$ curves with constant curvature on any Riemannian surface

Construction of examples showing regularity cannot be improved

Curves are smooth except at one point where curvature jumps

Abstract

We show that on any Riemannian surface for each $0<c<\infty$ there exists an immersed $C^{1,1}$ curve that is smooth and with curvature equal to $\pm c$ away from a point. We give examples showing that, in general, the regularity of the curve obtained by our procedure cannot be improved.