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

We show that on any Riemannian surface for each $0<c<\mathrm{\infty }$ 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.