The nearby Lagrangian conjecture for pinwheels

TL;DR

This paper proves the nearby Lagrangian conjecture for a class of singular Lagrangians called pinwheels in rational homology balls, using neck-stretching, symplectic blow-up, and symplectomorphism computations.

Contribution

It establishes the conjecture for pinwheels, introduces the pintwist symplectomorphism, and provides applications like non-squeezing and Lagrangian unknotting.

Findings

Any two embeddings of Lagrangian pinwheels are Hamiltonian isotopic.

The symplectomorphism group is generated by the pintwist.

Applications include non-squeezing, unknotting, and classification of pinwheels.

Abstract

The Lagrangian skeleton of the rational homology ball $B_{p,q}$, for $0<q<p$ coprime integers, is an immersed but not embedded Lagrangian, called a $(p,q)$-pinwheel. We show that any two embeddings of Lagrangian $(p,q)$-pinwheels in $B_{p,q}$ are related by a compactly supported Hamiltonian isotopy, establishing Arnold's nearby Lagrangian conjecture for this wide class of singular Lagrangians. Our proof has two largely independent parts: the first uses neck-stretching and the symplectic rational blow-up to understand embeddings of pinwheels up to symplectomorphism; the second computes that $\text{Symp}_c(B_{p,q})$ is generated by a twist about the pinwheel, which we call the pintwist $\tau_{p,q}$. We provide three applications of our methods: Gromov non-squeezing for pin-balls; a new proof of the local Lagrangian unknotting theorem of Eliashberg--Polterovich; and that the only Lagrangian $(n,m)$-pinwheel in $B_{p,q}$ is of type $(p,q)$.