High-dimensional families of holomorphic curves and three-dimensional energy surfaces

Let H: R⁴ R be any smooth function. This article introduces some arguments for extracting dynamical information about the Hamiltonian flow of H from high-dimensional families of closed holomorphic curves. We work in a very general setting, without imposing convexity or contact-type assumptions. For any compact regular level set Y, we prove that the Hamiltonian flow admits an infinite family of pairwise distinct, proper, compact invariant subsets whose union is dense in Y. This is a generalization of the Fish-Hofer theorem, which showed that Y has at least one proper compact invariant subset. We then establish a global Le Calvez-Yoccoz property for almost every compact regular level set Y: any compact invariant subset containing all closed orbits is either equal to Y or is not locally maximal. Next, we prove quantitative versions, in four dimensions, of the celebrated almost-existence theorem for Hamiltonian systems; such questions have been open for general Hamiltonians since the late 1980s. We prove that almost every compact regular level set of H contains at least two closed orbits, a sharp lower bound. Under explicit and C^-generic conditions on H, we prove almost-existence of infinitely many closed orbits.