A Viscosity method for the min-max construction of closed geodesics∗
Department of Mathematics, ETH Zentrum, 8093 Zürich, Switzerland.
Received: 6 June 2016
Accepted: 7 June 2016
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 ε-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.
Mathematics Subject Classification: 49J35 / 58B20 / 58E10
Key words: Geodesics / minimax problems / Finsler geometry
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