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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.

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    $\begingroup$ Probably you will have issues with uniqueness: compare en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem with en.wikipedia.org/wiki/Peano_existence_theorem.. ODE uniqueness can fail as soon as you drop Lipschitz! $\endgroup$ Nov 2, 2013 at 20:17
  • $\begingroup$ First, I think that the ODE generating the flow on the unit tangent bundle of $S$ is actually Lipschitz (even smooth) with respect to that variable, while it is only $L^1$ with respect to time. A useful reference might be the book "Shapes and diffeomorphisms" by Younes (2010), section 8.2 where the flow is studied of vector fields that are $C^k$ in space, but $L^2$ in time. With these ideas, I think one should be able to prove that the flow is unique and $c'(s)$ absolutely continuous. $\endgroup$ Nov 3, 2013 at 0:03

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I recommend to look in the book MR1117220 Alexandrov, A. D.; Reshetnyak, Yu. G. General theory of irregular curves.

It deals with curves in $R^n$, but I suppose that when your surface is smooth, it must be similar. For the case when your surface itself is not smooth, look to MR1263964 Reshetnyak, Yu. G. Two-dimensional manifolds of bounded curvature. Geometry, IV, 3–163, 245–250, Encyclopaedia Math. Sci., 70, Springer, Berlin, 1993.

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