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

Weonmo Lee (Department of Mathematics, POSTECH, and IBS Center for Geometry and Physics, Pohang, South Korea)

Yong-Geun Oh (Department of Mathematics, POSTECH, and IBS Center for Geometry and Physics, Pohang, South Korea)

Renato Vianna (Institute of Mathematics, Federal University of Rio de Janeiro, Brazil)

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 $\mathbb{C}P^2$.

Full Text (PDF format)

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