*Accepted Paper*

**Inserted:** 19 jul 2013

**Last Updated:** 19 jul 2013

**Journal:** Comm. Pure Appl. Math.

**Year:** 2013

**Abstract:**

Given a Tonelli Hamiltonian $H:T^*M \rightarrow \R$ of class $C^k$, with $k\geq 4$, we prove the following results:
(1) Assume there is a critical viscosity subsolution which is of class $C^{k+1}$ in
an open neighborhood of a positive orbit of a recurrent point of the projected Aubry set. Then,
there exists a potential $V:M \rightarrow \R$ of class $C^{k-1}$, small in $C^2$ topology, for which the Aubry set of the new Hamiltonian $H+V$
is either an equilibrium point or a periodic orbit. (2) For every $\epsilon>0$ there exists a potential $V:M \rightarrow \R$ of class $C^{k-2}$, with $\

V\

_{C^1} < \epsilon$, for which the Aubry set of the new Hamiltonian $H+V$
is either an equilibrium point or a periodic orbit. The latter result solves in the affirmative the Ma\~n\'e density conjecture in $C^1$ topology.

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