# Asymptotic behavior of Vianna's exotic Lagrangian tori $T_{a,b,c}$ in $\mathbb{CP}^2$ as $a+b+c \to \infty$

## TL;DR

This paper investigates the asymptotic properties of a family of monotone Lagrangian tori in complex projective plane, revealing bounds on their complements and their non-dense distribution as parameters grow large.

## Contribution

It establishes lower bounds on Gromov capacity of complements and shows the family of tori does not become dense in the space as parameters tend to infinity.

## Key findings

- Gromov capacity of the complement is at least one-third of the line area.
- Existence of a representative torus missing a nonzero size metric ball.
- The union of tori is not dense in P^2.

## Abstract

In this paper, we study various asymptotic behavior of the infinite family of monotone Lagrangian tori $T_{a,b,c}$ in $\mathbb{CP}^2$ associated to Markov triples $(a,b,c)$ described in \cite{Vi14}. We first prove that the Gromov capacity of the complement $\mathbb{CP}^2 \setminus T_{a,b,c}$ is greater than or equal to $\frac13$ of the area of the complex line for all Markov triple $(a,b,c)$. We then prove that there is a representative of the family $\{T_{a,b,c}\}$ whose loci completely miss a metric ball of nonzero size and in particular the loci of the union of the family is not dense in $\mathbb{CP}^2$.