Suppose that $S$ is a smooth complete surface, and $c\colon [0,L]\to S$ is a smooth curve in $S$, parametrized by arc-length. Then $c$ is uniquely determined by its initial tangent vector and its geodesic curvature $\kappa_g$.
I would like to relax the hypothesis that $\kappa_g$ is smooth: If $\kappa_g\colon [0,L]\to \mathbb{R}$ is only of class $L^1$ and $c'(0)$ is given, does there exist a unique curve in $S$, parametrized by arc-length, whose geodesic curvature is $\kappa_g$? More importantly, does it follow that $c'(s)$ is an absolutely continuous function of $s$?
To put it differently, is there an analogue of continuous dependence on the parameters of a differential equation when we are dealing with $L^1$ functions, at least for the differential equation (in the unit tangent bundle of $S$) which defines a curve in terms of its geodesic curvature?
I believe that the answer to both questions is yes, but I am unable to find a reference for this. I would greatly appreciate it if someone could provide one.
Thanks in advance.