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 $\mathbb{C}P^2$.

中文翻译：

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$\mathbb{C}P^2$ 中奇异拉格朗日环面 $T_{a,b,c}$ 的渐近行为为 $a+b+c \to \infty$

在本文中，我们研究了与马尔可夫三元组 $(a,b,c) 相关的 $\mathbb{C}P^2$ 中单调拉格朗日环面 $T_{a,b,c}$ 的无穷大族的各种渐近行为$ 在 [Via16] 中描述。我们首先证明补集 $\mathbb{C}P^2 \setminus T_{a,b,c}$ 的 Gromov 容量大于或等于 $\frac{1}{3}$ 的面积所有马尔可夫三元组 $(a,b,c)$ 的复数线。然后我们证明存在家族 $\lbrace T_{a,b,c} \rbrace$ 的代表，其轨迹完全错过了一个非零尺寸的度量球，特别是家族的并集轨迹在$\mathbb{C}P^2$。