A counterexample to the generalized Viterbo conjecture on the spectral distance in a disc cotangent bundle

Abstract

We prove the unboundedness of the spectral distance on the space of Lagrangians Hamiltonian isotopic to the zero section within the disc cotangent bundle of a closed manifold satisfying some topological assumptions. This disproves the generalization of a conjecture due to Claude Viterbo. The proof relies on a direct computation of the spectral distance between the zero section and its image by a magnetic Hamiltonian diffeomorphism via generating functions. We also classify all surfaces for which the conjecture holds over $\mathbb{Z}_2$ via a Floer-theoretic argument for covering spaces.