# Smoothness of closed geodesics

I'm trying to understand a paper about minimal surfaces and I came across the following problem: let $$N$$ be a smooth n-dimensional manifold embedded in an Euclidean space. Let $$\gamma$$ be a $$C^1$$ map from $$S^1$$ to $$N$$ which is a critical point of the Dirichlet energy $$\int_{S^1}\mid\nabla \gamma\mid^2dx$$. Does this imply that $$\gamma$$ is smooth?

Edit: actually my problem was a bit more general: the map $$\gamma$$ was obtained as the limit in $$C^0$$ of a minimizing sequence in $$C^\infty$$ for the Dirichlet energy. I could show that $$\gamma \in W^{1,2}$$, and since $$S^1$$ is 1-dimensional, $$\gamma$$ is a.e. differentiable (but the derivative has not to be continuous). Now I would like to show that $$\gamma$$ satisfy the geodesic equation $$\nabla_\gamma \dot{\gamma}=0$$, but for this I need to make sense of the second derivative of $$\gamma$$.

• It would be helpful if you provided more background and specified what you are having a hard time with. For example, if $\gamma$ were of class $C^2$, would you know the answer? – Amitai Yuval Sep 20 '18 at 18:00
• @AmitaiYuval Thank you for your answer! Yes, I think $C^2$ would be enough, since then we could use unicity of the solution of the geodesic differential equation for given initial data, but I have no idea how to prove that $\gamma \in C^2$. – tigro Sep 20 '18 at 20:47