TITLE: Regularity of geodesics
QUESTION [13 upvotes]: If $M$ is a graph of a $C^1$ function $f:\mathbb{R}^n\to\mathbb{R}$, is it true that the length minimizing geodesics on $M$ are $C^1$? I expect a counterexample.
For a related discussion see Metric angles in Riemannian manifolds of low regularity and Existence and uniqueness of geodesics in low regularity.

REPLY [3 votes]: A comment: a length minimising geodesic $(u, f(u))$ on $M:={\rm graph}(f)$, which it is convenient to give by its projection $u:[0,1]\to\mathbb{R}^n $ on the domain of $f$,  can be identified with a critical point $u\in H^1([0,1],\mathbb{R}^n)$ of $$E(u):={1\over 2}\int^1_0 \|\dot u \|^2+\big(\nabla f(u )\cdot \dot u \big)^2 dt=  {1\over 2}\int^1_0 \|\dot u \|^2+\big(\partial_t f(u )\big)^2 dt.$$
There could be a simple argument to show regularity of the minimisers.