Infinitely many exotic monotone Lagrangian tori in CP^2

TL;DR

This paper constructs infinitely many distinct monotone Lagrangian tori in complex projective plane CP^2, using symplectic field theory to distinguish them up to Hamiltonian isotopy, linked to Markov triples.

Contribution

It introduces a new family of monotone Lagrangian tori in CP^2 associated with Markov triples and proves their non-Hamiltonian isotopy using symplectic field theory.

Findings

Constructed infinitely many monotone Lagrangian tori in CP^2.

Proved these tori are not Hamiltonian isotopic to each other.

Connected the construction to Markov triples.

Abstract

Related to each degeneration from CP^2 to CP(a^2,b^2,c^2), for (a,b,c) a Markov triple - positive integers satisfying a^2 + b^2 + c^2 = 3abc - there is a monotone Lagrangian torus, which we call T(a^2,b^2,c^2). We employ techniques from symplectic field theory to prove that no two of them are Hamiltonian isotopic to each other.