The Dynamics on Three-Dimensional Strictly Convex Energy Surfaces

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We show that a Hamiltonian flow on a three-dimensional strictly convex energy surface S C R4 possesses a global surface of section of disc type. It follows, in particular, that the number of its periodic orbits is either 2 or oc, by a recent result of J. Franks on area-preserving homeomorphisms of an open annulus in the plane. The construction of this surface of section is based on partial differential equations of Cauchy-Riemann type for maps from punctured Riemann surfaces into R x S3 equipped with special almost complex structures.