The dynamics on three-dimensional strictly convex energy surfaces


We show that a Hamiltonian flow on a three-dimensional strictly convex energy surface $S\subset \mathbb{R}^4$ possesses a global surface of section of disc type. It follows, in particular, that the number of its periodic orbits is either $2$ or $\infty$, 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 $\mathbb{R}\times S^3$ equipped with special almost complex structures.


Helmut Hofer

Krzysztof Wysocki

Eduard Zehnder