zbMATH — the first resource for mathematics

A complete proof of Viterbo’s uniqueness theorem on generating functions. (English) Zbl 0952.53037
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 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 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.

53D12 Lagrangian submanifolds; Maslov index
57R52 Isotopy in differential topology
53D05 Symplectic manifolds, general
Full Text: DOI