On exotic Lagrangian tori in \(\mathbb{CP}^{2}\)
Before the paper under review, one only knew two monotone Lagrangian torus in \(\mathbb{CP}^2\) endowed with the standard Fubini-Study form up to Hamiltonian isotopy, the so-called Clifford and Chekanov tori. The first is obtained by symplectically embedding the products \((S^1(r))^n\subset\mathbb{CP}^2\) into \(\mathbb{CP}^2\), and the second was introduced by \textit{Yu. V. Chekanov} [Math. Z. 223, No. 4, 547--559 (1996; Zbl 0877.58024)] and is not Hamiltonian isotopic to the former.NEWLINENEWLINE\textit{D. Auroux} studied the SYZ mirror dual of a singular special Lagrangian torus fibration given on the complement of an anticanonical divisor in \(\mathbb{CP}^2\) [J. Gökova Geom. Topol. GGT 1, 51--91 (2007; Zbl 1181.53076)] and found this fibration to interpolate between the Clifford torus and a slightly modified version of the Chekanov torus described by \textit{Y. Eliashberg} and \textit{L. Polterovich} [AMS/IP Stud. Adv. Math. 2, 313--327 (1997; Zbl 0889.57036)].NEWLINENEWLINEA degeneration of \(\mathbb{CP}^2\) to the weighted projective space \(\mathbb{CP}(a^2,b^2,c^2)\) for a Markov triple \((a,b,c)\), - \(a + b + c = 3abc\), can be illustrated with almost toric fibrations defined by \textit{M. Symington} [Proc. Symp. Pure Math. 71, 153--208 (2003; Zbl 1049.57016)]. Such a degeneration yields a monotone Lagrangian torus \(T(a^2,b^2,c^2)\) in \(\mathbb{CP}^2\), which is the Clifford torus for \((a,b,c)=(1,1,1)\), and Chekanov torus for \((a,b,c)=(1,1,4)\).NEWLINENEWLINEIn the paper under review, by reinterpreting Auroux's construction using almost toric fibrations the author showed: (i) the monotone \(T(1,4,25)\) torus is not symplectomorphic to either the monotone Clifford torus or the monotone Chekanov torus (in Corollary 6.5), and hence is not Hamiltonian isotopic to the Clifford or Chekanov torus, (ii) some tori of type \(T(1,4,25)\) is nondisplaceable (in Corollary 6.7).NEWLINENEWLINEIn addition, using techniques from symplectic field theory the author recently proved that monotone Lagrangian tori \(T(a^2,b^2,c^2)\) are mutually not Hamiltonian isotopic to each other [``Infinitely many exotic monotone Lagrangian tori in \(\mathbb{CP}^2\)'', Preprint, \url{arXiv:1409.2850}].NEWLINENEWLINEThe paper is very nicely written and contains many figures.
scientific article; zbMATH DE number 6356612
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