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Infinitely many Lagrangian fillings

Roger Casals University of California, Davis CA Honghao Gao Michigan State University, East Lansing MI

Differential Geometry mathscidoc:2203.10008

Annals of Mathematics, 195, 207-249, 2022.1
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 that induce faithful actions of the modular group PSL(2,Z) and the mapping class group M0,4 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.
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@inproceedings{roger2022infinitely,
  title={Infinitely many Lagrangian fillings},
  author={Roger Casals, and Honghao Gao},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220317161312774154997},
  booktitle={Annals of Mathematics},
  volume={195},
  pages={207-249},
  year={2022},
}
Roger Casals, and Honghao Gao. Infinitely many Lagrangian fillings. 2022. Vol. 195. In Annals of Mathematics. pp.207-249. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220317161312774154997.
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