Non-isotopic monotone Lagrangian submanifolds of $\mathbb{C}^n$

TL;DR

This paper explores the construction and classification of monotone and non-monotone Lagrangian submanifolds in complex Euclidean spaces, linking their properties to Delzant polytopes and introducing new embeddings with distinct Hamiltonian isotopy classes.

Contribution

It establishes a connection between Delzant polytopes and monotone Lagrangian submanifolds, introduces the Lagrangian Delzant Theorem, and constructs multiple non-isotopic embeddings of specific manifolds into complex space.

Findings

Monotone Lagrangians correspond to Fano polytopes.

Constructed multiple non-Hamiltonian isotopic embeddings of spheres and tori.

Established a Lagrangian analogue of Delzant's theorem.

Abstract

Let $P$ be a Delzant polytope in $\mathbb{R}^k$ with $n+k$ facets. We associate a closed Lagrangian submanifold $L$ of $\mathbb{C}^n$ to each Delzant polytope. We prove that $L$ is monotone if and only if and only if the polytope $P$ is Fano. We pose the "Lagrangian version of Delzant Theorem". Then for even $p$ and $n$ we construct $\frac{p}{2}$ monotone Lagrangian embeddings of $S^{p-1} \times S^{n-p-1} \times T^2$ into $\mathbb{C}^n$, no two of which are related by Hamiltonian isotopies. Some of these embeddings are smoothly isotopic and have equal minimal Maslov numbers, but they are not Hamiltonian isotopic. Also, we construct infinitely many non-monotone Lagrangian embeddings of $S^{2p-1} \times S^{2p-1} \times T^2$ into $\mathbb{C}^{4p}$, no two of which are related by Hamiltonian isotopies.