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arXiv · 1904.11775

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

Weonmo Lee·Yong-Geun Oh·Renato Vianna
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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 $\mathbb{CP}^2$.

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Weonmo Lee, Yong-Geun Oh, Renato Vianna. 2019-04-26. Asymptotic behavior of Vianna's exotic Lagrangian tori $T_{a,b,c}$ in $\mathbb{CP}^2$ as $a+b+c \to \infty$. https://arxiv.org/abs/1904.11775

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