Proof of the $C^2$ Ma\~n\'e's conjecture on surfaces

We prove that $C^2$ generic hyperbolic Ma\~n\'e sets contain a periodic periodic orbit. In dimension 2, adding a result by Contreras, Figalli, Rifford, which states that $C^2$ generic Ma\~n\'e sets are hyperbolic; we obtain Ma\~n\'e's Conjecture for surfaces in the $C^2$ topology: Given a Tonelli Lagrangian $L$ on a compact surface $M$ there is a $C^2$ open and dense set of functions $f:M\to\mathbb{R}$ such that the Ma\~n\'e set of the Lagrangian $L+f$ is a hyperbolic periodic orbit.