Ergodic Theory and Dynamical Systems

Table of Contents - 2004 - Volume 24, Issue 06  


On variational and topological properties of C1 invariant Lagrangian tori

M. J. DIAS CARNEIRO a1 and RAFAEL O. RUGGIERO a2
a1 Universidade Federal de Minas Gerais, Dep. de Matemática, ICEx, Belo Horizonte, MG, Brasil (e-mail: carneiro@mat.ufmg.br)
a2 Pontificia Universidade Católica do Rio de Janeiro, PUC-Rio, Dep. de Matemática, Rua Marqués de São Vicente 225, Gávea, Rio de Janeiro, Brasil (e-mail: rorr@mat.puc-rio.br)

Article author query
dias carneiro m   [Google Scholar] 
ruggiero r   [Google Scholar] 
 

Abstract

We study C1 invariant, two-dimensional tori of constant energy without singularities of the Euler–Lagrange flow of a convex, superlinear Lagrangian defined in the tangent space of a closed surface. We show that in regular energy levels, such tori are graphs of the canonical projection if and only if they are minimizing. This result was known for C3 invariant, Lagrangian tori of Finsler Lagrangians, which implied the statement for C3 invariant Lagrangian tori of unimodal Lagrangians with a high-energy level. We also show that graphs have energy $E \geq c_0(L)$, where c0(L) is Mañé's strict critical value; and that the presence of Reeb components in such tori implies that the energy is c0(L). Moreover, if we assume that the Lagrangian is unimodal, we show that Lagrangian invariant tori with no singularities have no Reeb components, regardless of the energy level.

(Received December 3 2001)
(Revised December 13 2002)