Generic uniqueness of shortest closed geodesics (Q343008)

It is well known that if \(F\) is a Finsler metric on a closed manifold \(M\), then in each homotopy class of free loops there exists at least one loop minimizing the associated length functional, which is thus a geodesic. See, e.g., Theorem 8.7.1(2) in [\textit{D. Bao} et al., An introduction to Riemann-Finsler geometry. New York, NY: Springer (2000; Zbl 0954.53001)]. However, as the flat torus clearly shows, in general these minimizers are not unique. In this paper the author shows that for conformally generic perturbations of the Finsler metric, they become unique. The notion of genericity used here is based on a kind of geometric perturbation related, via Maupertuis' principle, to the ones introduced for Lagrangians in [\textit{R. Mañé}, Nonlinearity 9, No. 2, 273--310 (1996; Zbl 0886.58037)]. The proof of the main theorem is based on an abstract result on the minimization of varying functionals, which is obtained following the ideas of R. Mañé. The author considers possible applications of the ideas in this paper to genericity problems associated to integral currents in geometric measure theory.