Weonmo Lee, Yong-Geun Oh, Renato Vianna

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 $\mathbb{C}P^2$.