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

TL;DR

This paper investigates the topology of monotone Lagrangian submanifolds in complex Euclidean spaces, establishing restrictions on their topology, especially in dimension three, and proving an h-principle for their immersions.

Contribution

It provides new topological restrictions for monotone Lagrangians in $ extbf{C}^n$, including a classification result for orientable 3-manifolds, and introduces an h-principle for their immersions.

Findings

Orientable 3-manifolds can only embed as monotone Lagrangians in $ extbf{C}^3$ if they are products.

Derived restrictions on the topology of monotone Lagrangians using moduli space analysis.

Proved an h-principle for monotone Lagrangian immersions.

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.