QUALITATIVE THEORY OF DYNAMICAL SYSTEMS 4, 439–454 (2004) ARTICLE NO. 71 Convex Energy Levels of Hamiltonian Systems Pedro A. S. Salom˜ao * Instituto de Matem´atica e Estat´ ıstica, Universidade de S˜ao Paulo, Rua do Mat˜ao 1010, Cidade Universit´aria, 05508-090 S˜aoPaulo, SP, Brazil E-mail: psalomao@ime.usp.br Submitted: July 27, 2003 Accepted: March 7, 2004 To Professor Jorge Sotomayor for his 60 th birthday We give a simple necessary and sufficient condition for a non-regular en- ergy level of a Hamiltonian system to be strictly convex. We suppose that the Hamiltonian function is given by kinetic plus potential energy. We also show that this condition holds for several Hamiltonian functions, including the H´ enon-Heiles one. Key Words: Hamiltonian systems, convexity, positive curvature, saddle-center, H´ enon-Heiles Hamiltonian. 1. INTRODUCTION An important theorem due to Hofer, Zehnder and Wysocki [11] for Hamiltonian systems with two degrees of freedom states that a strictly convex energy level diffeomorphic to S 3 always has a periodic orbit which is the boundary of a global surface of section of disk-type. Moreover, this implies the existence of 2 or infinitely many periodic orbits in that en- ergy level. This geometric property turns out to be of great importance to understand the topological structure of its orbits. When the Hamiltonian function is of the form p 2 x +p 2 y 2 + V (x, y) then the conditions of convexity can be expressed in terms of the potential function V . In this paper, we give these geometric conditions in the regular case * partially supported by FAPESP 439