The Legendrian Hopf Link has exactly two Lagrangian fillings

TL;DR

This paper proves that the standard Legendrian Hopf link has exactly two embedded exact Lagrangian fillings, classifying all such fillings using a neck-stretching technique and pseudoholomorphic fibrations.

Contribution

It establishes the uniqueness of the two known Lagrangian fillings of the Legendrian Hopf link by employing a novel neck-stretching and pseudoholomorphic fibration approach.

Findings

Exactly two embedded exact Lagrangian fillings exist for the Legendrian Hopf link.

All Lagrangian fillings are Hamiltonian isotopic to one of the two known fillings.

The method constructs explicit isotopies via pseudoholomorphic fibrations.

Abstract

We prove that there are precisely two embedded exact Lagrangian fillings of the standard Legendrian Hopf link, up to compactly supported Hamiltonian isotopy. It was known that the standard Legendrian Hopf link admitted at least two such Lagrangian fillings: we show these are all. Specifically, we use a type of neck-stretching procedure to construct a pseudoholomorphic conic fibration that makes a given arbitrary exact Lagrangian filling fiber over a real curve, under a global pseudoholomorphic Lefschetz fibration. This then allows for an explicit Hamiltonian isotopy to be constructed from any given Lagrangian filling to one of two known standard fillings.