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

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.