A Viscosity method for the min-max construction of closed geodesics

Michelat, Alexis; Rivière, Tristan

ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 4, pp. 1282-1324.

We present a viscosity approach to the min-max construction of closed geodesics on compact Riemannian manifolds of arbitrary dimension. The existence is proved in the case of surfaces, and reduced to a topological condition in general. We also construct counter-examples in dimension $1$ and $2$ to the $\epsilon$-regularity in the convergence procedure. Furthermore, we prove the lower semi-continuity of the index of our sequence of critical points converging towards a closed non-trivial geodesic.

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