A complete proof of Viterbo's uniqueness theorem on generating functions
type: Item
instance of: scholarly article
title: A complete proof of Viterbo's uniqueness theorem on generating functions
zbMATH Open document ID: 0952.53037
DOI: 10.1016/S0166-8641(98)00049-2
author: David Théret
published in: Topology and its Applications
publication date: 2000-04-09T00:00:00Z
review text: Let \( M \) be an \(n\)-dimensional connected closed smooth manifold, \( T ^\star M \) its cotangent bundle, \( L \) a closed Lagrangian submanifold of \( T ^\star M , \) and \( (\varphi _t) _{t \in [0,1]} \) a Hamiltonian isotopy of \( T ^\star M. \) If \( L \) admits a generating and quadratic at infinity (abbreviated gqi) function, then \( \varphi _1 (L) \) also has a gqi function [see \textit{J.-C. Sikorav}, Comment. Math. Helv. 62, 62-73 (1987; Zbl 0684.58015)]. Since the zero section \( M \) obviously has a gqi function, it follows that \( \varphi _1 (M) \) always has a gqi function. Viterbo's uniqueness theorem asserts that the gqi functions of \( \varphi _1 (M) \) are all equivalent [see \textit{Claude Viterbo}, Math. Ann. 292, 685-710 (1992; Zbl 0780.58023)], that is any two of them can be made equal after a succession of basic operations: addition of a constant, diffeomorphism operation, stabilization. The author's motivation for the present paper is that the initial proof of this uniqueness result was a little too elliptic to be fully convincing for many readers. He has thus reworked every step, which lead him to change some parts -- in particular in what he calls the ``invariance of the uniqueness property under isotopies'' because an incorrect use of Sikorav's paper (see loc. cit.) was made in the original proof. He also proves that both Sikorav's existence and Viterbo's uniqueness theorems can be generalized to symplectic isotopies, using generating forms instead of functions.
reviewed by: Mircea Craioveanu
Mathematics Subject Classification ID: 53D12
Mathematics Subject Classification ID: 57R52
Mathematics Subject Classification ID: 53D05
zbMATH DE Number: 1340387
zbMATH Keywords: Lagrangian submanifold
zbMATH Keywords: Hamiltonian isotopy
zbMATH Keywords: symplectic isotopy
zbMATH Keywords: generating function
zbMATH Keywords: quadratic at infinity
zbMATH Keywords: generating form
MaRDI profile type: Publication
full work available at URL: s0166-8641(98)00049-2
OpenAlex ID: W2152021108
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description: scientific article; zbMATH DE number 1340387