Existence of generating families on Lagrangian cobordisms
Generating families is a powerful tool that can be used to study Legendrian submanifolds and Lagrangian cobordisms between such. However, the existence and construction of a generating family is in general very difficult. A homotopical necessary condition for the existence of a generating family is that the stable Lagrangian Gauss map is null-homotopic. For Legendrian links, the stable Lagrangian Gauss map is null-homotopic if and only if the Maslov class is trivial.\N\NThe main result of the paper under review is the following: Let \(L \subset J^1M \times \mathbb{R}_{>0}\) be an exact Lagrangian cobordism from \(\Lambda_- \subset J^1M\) to \(\Lambda_- \subset J^1M\). A generating family linear at infinity of \(\Lambda_-\) extends to \(L\) (up to stabilizations) if and only if the stable Lagrangian Gauss map \(L \to U/O\) is null-homotopic compatibly with the null-homotopy of the stable Lagrangian Gauss map determined by the generating family of \(\Lambda_-\). In particular, this implies that there exists a generating family linear at infinity for \(\Lambda_+\).
scientific article; zbMATH DE number 7940265
scientific article