# Zoll contact forms are local maximisers of the systolic ratio

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Alberto Abbondandolo, Ruhr-Universität Bochum

A central question from systolic geometry is to find upper bounds for the systolic ratio of a Riemannian metric on a closed $n$-dimensional manifold, i.e. the ratio of the $n$-th power of the shortest length of closed geodesics by the volume. This question can be naturally extended to Reeb flows, a class of dynamical systems including geodesic flows and induced by a contact form on a closed manifold. The aim of this talk is to discuss a recent result obtained in collaboration with Gabriele Benedetti: Zoll contact forms, i.e. forms such that all the orbits of the induced Reeb flow are periodic with the same period, are local maximisers of the systolic ratio. Consequences of this result are: (i) sharp systolic inequalities for Riemannian and Finsler metrics close to Zoll ones, (ii) the perturbative case of a conjecture of Viterbo on the symplectic capacity of convex bodies, (iii) a generalization of Gromov's non-squeezing theorem in the intermediate dimensions for symplectomorphisms that are close to linear ones.