Asymptotic behavior of exotic Lagrangian tori \(T_{a,b,c}\) in \(\mathbb{C}P^2\) as \(a+b+c\to\infty\) (Q2048919)

The article under review is concerned with the asymptotic behavior of a certain family of monotone Lagrangian tori in \(\mathbb CP^2\). These Lagrangian tori are constructed in a previous work from [\textit{R. Ferreira de Velloso Vianna}, J. Topol. 9, No. 2, 535--551 (2016; Zbl 1350.53102)] and denoted by \(T_{a,b,c}\) where \((a,b,c)\) is a Markov triple satisfying \(a^2 + b^2 + c^2 = 3abc\). A deep result of \(T_{a,b,c}\) is that they are not pairwise Hamiltonian isotopic to each other. Therefore, it is a curious direction to investigate the geometric behavior of these \(T_{a,b,c}\) in \(\mathbb CP^2\) when \(a,b,c \to +\infty\). Here, \(\mathbb CP^2\) is associated with the standard Fubini-Study form \(\omega_{\mathrm{FS}}\) such that the area of the complex line is \(2\pi\). There are two main results in this article towards this direction. Consider the following quantity as a symplectic invariant, \[ c_G(\mathbb CP^2; T_{a,b,c}) : = \sup_{e}\{\pi r^2 \,| \, e: B^4(r) \to \mathbb CP^2 \backslash T_{a,b,c} \, \mbox{is a symplectic embedding}\}, \] which is called the relative Gromov area of \(T_{a,b,c}\) in \(\mathbb CP^2\). Note that a similar but different symplectic invariant is studied by \textit{P. Biran} and \textit{O. Cornea} in [Geom. Topol. 13, No. 5, 2881--2989 (2009; Zbl 1180.53078)]. Then the first main result is that \[ \inf_{a,b,c \to +\infty} c_G(\mathbb CP^2; T_{a,b,c}) \geq \frac{2\pi}{3} \] and it is conjectured that the above inequality is an equality. This result is not strong enough to conclude that the union of all \(T_{a,b,c}\) is not dense in \(\mathbb CP^2\), since for different \(T_{a,b,c}\) it might need different symplectic embeddings of \(B^4(r)\) to obtain the estimation above. However, the article indeed confirms that the union of certain Hamiltonian isotopic images of \(T_{a,b,c}\) is not dense in \(\mathbb CP^2\), via the geometric mutation theorem of the Lagrangian seeds in [\textit{V. Shende} et al., ``On the combinatorics of exact Lagrangian surfaces'', Preprint, \url{arXiv:1603.07449}]. The second main result in this article investigates the size of a Weinstein neighborhood of \(T_{a,b,c}\) in \(\mathbb CP^2\). In general, if \(L \subset M\) is a compact Lagrangian submanifold equipped with a metric \(g\), denote by \(\pi: T^*L \to L\) the canonical projection and define \[ \mathfrak w_{\mathrm{DW}}(L; M) = \sup_{\Phi}\left\{ \inf_{q \in L} \left(\inf_{x \in \pi^{-1}(q) \cap \partial \mathcal U} \|\Phi(x)\|_{g(\pi(\Phi(x)))} \right)\right\} \] where the supremum is taken over all the Darboux-Weinstein charts \(\Phi: \mathcal U \to \mathcal V\) of \(L\) in \(M\). Then the result proves that \[ \inf_{a,b,c \to +\infty} \mathfrak w_{\mathrm{DW}}(T_{a,b,c}; \mathbb CP^2) =0. \] Here, we view \(T_{a,b,c}\) as the image of Lagrangian embeddings from \(T^2\) with a fixed Riemannian metric. This illustrates a wild asymptotic behavior of \(T_{a,b,c}\), and its proof is based on the star-flux theory developed in [\textit{E. Shelukhin} et al., ``Geometry of symplectic flux and Lagrangian torus fibrations'', Preprint, \url{arXiv:1804.02044}] In addition, this article also studies the ball packing problem in \(\mathbb CP^2 \backslash T_{a,b,c}\), which, by the definition of the symplectic blowup, is equivalent to the possibility of Lagrangian embeddings of \(T_{a,b,c}\) into \(\mathbb CP^2 \# k \overline{\mathbb CP^2}\) for some \(k \geq 0\). A result in this article shows that this is possible for any \(T_{a,b,c}\) if \(k \leq 5\). This answers a question from [\textit{Y. Chekanov} and \textit{F. Schlenk}, Electron. Res. Announc. Math. Sci. 17, 104--121 (2010; Zbl 1201.53083)].