A complete proof of Viterbo's uniqueness theorem on generating functions

Théret, David

HAL, hal-01816521 / Harvested from HAL

Résumé

We present here the complete proof of a theorem of Claude Viterbo, stating a uniqueness property for quadratic at infinity generating functions. In his paper [27], Claude Viterbo stated a uniqueness result about the quadratic at infinity generating functions which generate a Lagrangian submanifold hamiltonianly isotopic to the zero section, in the cotangent bundle of a closed manifold. This enabled him to define capacities for the open sets in R 2n , with applications such as the non-squeezing theorem, the camel problem and the Weinstein conjecture. The motivation for the present text is that the initial proof was a little too elliptic to be fully convincing for many readers. We have thus reworked every step, which lead us to change some parts—in particular in what we call the " invariance of the uniqueness property under isotopies " (Section 5) because an incorrect use of Sikorav's paper [21] was made in the original proof. We also generalize both Sikorav's existence and Viterbo's uniqueness theorems to the non-exact case, using generating forms instead of functions. This text is organized as follows. In Section 1, we recall the relevant basic definitions of symplectic geometry. Section 2 is concerned with generating functions and forms. We state Viterbo's result and the generalization to the non-exact case in Section 3, and we give there a brief sketch of the proofs. The remaining sections are devoted to the complete proofs. 1

Détail

Publié le : 1999-07-04
Classification:  Hamiltonian and Lagrangian systems,  Symplectic geometry,  Differential topology,  Embeddings,  Isotopy,  Fiber bundles,  Manifolds of mappings AMS classification: 58F05,  55R10,  57R40,  57R52,  58D15,  [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG]