Mathematical Research Letters

Volume 21 (2014)

Number 6

Remarks on monotone Lagrangians in $\mathbf{C}^n$

Pages: 1241 – 1255

DOI: http://dx.doi.org/10.4310/MRL.2014.v21.n6.a2

Authors

Jonathan David Evans (Department of Mathematics, University College London, England)

Jarek Kedra (Institute of Mathematics, University of Aberdeen, Scotland; and Instytut Matematyki, Universytet Szczecinski, Szczecin, Poland)

Abstract

We derive some restrictions on the topology of a monotone Lagrangian submanifold $L \subset \mathbf{C}^n$ by making observations about the topology of the moduli space of Maslov 2 holomorphic discs with boundary on $L$ and then using Damian’s theorem which gives conditions under which the evaluation map from this moduli space to $L$ has nonzero degree. In particular, we prove that an orientable $3$-manifold admits a monotone Lagrangian embedding in $\mathbf{C}^3$ only if it is a product, which is a variation on a theorem of Fukaya. Finally, we prove an $h$-principle for monotone Lagrangian immersions.

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