In symplectic topology, subcritical isotropic submanifolds have receivedmuch less attention than their Lagrangian siblings. Until very recently, wediscovered that for any n; k 2 N with n > 1, there are two exact isotropicn-tori Tn;k and T0n;k in Cn+k which are smoothly isotopic but have distinctHamiltonian isotopy classes. We also applied this result to show that the treetypes of Lagrangian twist tori in C3 constructed by Chekanov-Schlenk andthe monotone Cli?ord torus in C3 are pairwise Hamiltonian non-isotopic asmonotone Lagrangian tori in C3. These results indicate that not only exactsubcritical isotropic submanifolds by themselves are a interesting topic ofstudy in symplectic topology, they also play a nontrivial role in problemsrelated to Lagrangian submanifolds.In this research project, we will continue with our study on exact isotropictori in Cm and their relations with monotone Lagrangian tori. Topics thatwe will focus on include (1) ?niteness of Hamiltonian isotopy classes of exactisotropic tori in Cm, (2) Hamiltonian isotopy rigidity of exact isotropic toriunder stabilizations of the ambient spaces, and (3) the relation between theHamiltonian isotopy class of a monotone Lagrangian torus and that of itsexact isotropic toric ?ber.Results from this proposed research will deepen our understanding ofLagrangian tori and exact isotropic tori in Cm, and may have applicationsto related problems in symplectic topology under more genera settings.
|Effective start/end date||1/08/16 → 30/11/17|
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