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# Journal of Symplectic Geometry

## Volume 19 (2021)

### Number 3

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

Pages: 607 – 634

DOI: https://dx.doi.org/10.4310/JSG.2021.v19.n3.a4

#### Authors

#### Abstract

In this paper, we study various asymptotic behavior of the infinite family of monotone Lagrangian tori $T_{a,b,c}$ in $\mathbb{C}P^2$ associated to Markov triples $(a,b,c)$ described in [Via16]. We first prove that the Gromov capacity of the complement $\mathbb{C}P^2 \setminus T_{a,b,c}$ is greater than or equal to $\frac{1}{3}$ 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 $\lbrace T_{a,b,c} \rbrace$ 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 $\mathbb{C}P^2$.

W.L. and Y.O. were supported by the IBS project IBS-R003-D1.

R.V. was supported by Brazil’s National Council of scientific and technological development CNPq, via the research fellowships 405379/2018-8 and 306439/2018-2, and by the Serrapilheira Institute grant Serra-R-1811-25965.

Received 8 May 2019

Accepted 9 November 2020

Published 21 July 2021