On the topology of monotone Lagrangian submanifolds

TL;DR

This paper establishes new topological obstructions for monotone Lagrangian submanifolds in complex Euclidean spaces, showing certain connected sums cannot admit such embeddings and characterizing all orientable Lagrangians in dimension three.

Contribution

It introduces novel topological obstructions based on homology of the universal cover, expanding understanding of which manifolds can be monotone Lagrangian submanifolds.

Findings

Nontrivial connected sums of odd-dimensional manifolds do not admit monotone Lagrangian embeddings.

In dimension three, the only orientable Lagrangians are products of a circle and a surface.

Some manifolds admit usual Lagrangian embeddings but not monotone ones.

Abstract

We find new obstructions on the topology of monotone Lagrangian submanifolds of $C^{n}$ under some hypothesis on the homology of their universal cover. In particular we show that nontrivial connected sums of manifolds of odd dimensions do not admit monotone Lagrangian embeddings into $\C^{n}$ whereas some of these examples are known to admit usual Lagrangian embeddings. In dimension three we get as a corollary that the only orientable Lagrangians in ${\bf C}^{3}$ are products $S^{1}\times \Sigma$.