Existence of generating families on Lagrangian cobordisms

TL;DR

This paper establishes conditions under which generating families extend across Lagrangian cobordisms, linking geometric properties with algebraic invariants, and providing criteria for the existence of generating families on fillings.

Contribution

It proves that generating families extend over Lagrangian cobordisms if formal obstructions vanish, offering new insights into the topology of Legendrian and Lagrangian submanifolds.

Findings

Generating families extend if and only if formal obstructions vanish.

Lagrangian fillings with trivial stable Gauss map admit generating families.

Extension of generating families is characterized by stabilization and obstruction vanishing.

Abstract

For an embedded exact Lagrangian cobordism between Legendrian submanifolds in the 1-jet bundle, we prove that a generating family linear at infinity on the Legendrian at the negative end extends to a generating family linear at infinity on the Lagrangian cobordism after stabilization if and only if the formal obstructions vanish. In particular, a Lagrangian filling with trivial stable Lagrangian Gauss map admits a generating family linear at infinity.