  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.
