Lagrangian metric geometry with Riemannian bounds

We study collections of exact Lagrangian submanifolds respecting some uniform Riemannian bounds, which we equip with a metric naturally arising in symplectic topology (e.g., the Lagrangian Hofer metric or the spectral metric).We exhibit many metric and symplectic properties of these spaces, such that they have compact completions and that they contain only finitely many Hamiltonian isotopy classes.We then use this to exclude many unusual phenomena from happening in these bounded spaces.Taking limits in the bounds, we also conclude that there are at most countably many Hamiltonian isotopy classes of exact Lagrangian submanifolds in a Liouville manifold.Under some mild topological assumptions, we get analogous results for monotone Lagrangian submanifolds with a fixed monotonicity constant.Finally, in the process of showing these results, we get new results on the Riemannian geometry of cotangent bundles and surfaces which might be of independent interest. .We will also denote by L ?1 the space of all Lagrangian submanifolds respecting ?.Moreover, the metric d that we consider will be a so-called Chekanov-type metric [11] and be bounded from above by the Lagrangian Hofer metric.In practice, one can think of d as one of the following metrics.d D d H This is the case of the Lagrangian Hofer metric, which is due to Chekanov [14].This is the case of the spectral metric, originally due to Viterbo [43] for Lagrangian submanifolds of T L Hamiltonian isotopic to the zero-section.The metric may be extended to all exact Lagrangian MSC2020: primary 53D12; secondary 51F99, 53C17, 53C22.