Infinitely many Lagrangian fillings


We prove that all maximal-tb positive Legendrian torus links $(n,m)$ in the standard contact 3-sphere, except for $(2,m)$, $(3,3),(3,4)$ and $(3,5)$, admit infinitely many Lagrangian fillings in the standard symplectic 4-ball. This is proven by constructing infinite order Lagrangian concordances which induce faithful actions of the modular group $\mathrm{PSL}(2,\mathbb{Z})$ and the mapping class group $M_{0,4}$ into the coordinate rings of algebraic varieties associated to Legendrian links. In particular, our results imply that there exist Lagrangian concordance monoids with subgroups of exponential-growth, and yield Stein surfaces homotopic to a 2-sphere with infinitely many distinct exact Lagrangian surfaces of higher-genus. We also show that there exist infinitely many satellite and hyperbolic knots with Legendrian representatives admitting infinitely many exact Lagrangian fillings.


Roger Casals

Department of Mathematics, University of California Davis, Davis, CA

Honghao Gao

Department of Mathematics, Michigan State University, East Lansing, MI