# American Institute of Mathematical Sciences

2010, 17: 104-121. doi: 10.3934/era.2010.17.104

## Notes on monotone Lagrangian twist tori

 1 Moscow Center for Continuous Mathematical Education, B. Vlasievsky per. 11, Moscow 121002, Russian Federation 2 Institut de Mathématiques, Université de Neuchâtel, Rue Émile Argand 11, CP 158, 2009 Neuchâtel, Switzerland

Received  April 2010 Revised  July 2010 Published  October 2010

We construct monotone Lagrangian tori in the standard symplectic vector space, in the complex projective space and in products of spheres. We explain how to classify these Lagrangian tori up to symplectomorphism and Hamiltonian isotopy, and how to show that they are not displaceable by Hamiltonian isotopies.
Citation: Yuri Chekanov, Felix Schlenk. Notes on monotone Lagrangian twist tori. Electronic Research Announcements, 2010, 17: 104-121. doi: 10.3934/era.2010.17.104
##### References:
 [1] P. Albers and U. Frauenfelder, A non-displaceable Lagrangian torus in $T^$*$S^2$, Comm. Pure Appl. Math., 61 (2008), 1046-1051. doi: doi:10.1002/cpa.20216.  Google Scholar [2] V. I. Arnold, On a characteristic class entering into conditions of quantization, Funkcional. Anal. i Prilozen., 1 (1967), 1-14. doi: doi:10.1007/BF01075861.  Google Scholar [3] D. Auroux, Mirror symmetry and T-duality in the complement of an anticanonical divisor, J. Gökova Geom. Topol., 1 (2007), 51-91.  Google Scholar [4] P. Biran and O. Cornea, A Lagrangian quantum homology, in "New Perspectives and Challenges in Symplectic Field Theory," CRM Proc. Lecture Notes 49, AMS, (2009), 1-44.  Google Scholar [5] P. Biran and O. Cornea, Rigidity and uniruling for Lagrangian submanifolds, Geom. Topol., 13 (2009), 2881-2989. doi: doi:10.2140/gt.2009.13.2881.  Google Scholar [6] Yu. V. Chekanov, Lagrangian tori in a symplectic vector space and global symplectomorphisms, Math. Z., 223 (1996), 547-559.  Google Scholar [7] Yu. V. Chekanov, Lagrangian intersections, symplectic energy, and areas of holomorphic curves, Duke Math. J., 95 (1998), 213-226. doi: doi:10.1215/S0012-7094-98-09506-0.  Google Scholar [8] Yu. Chekanov and F. Schlenk, Twist tori I: Construction and classification,, in preparation., ().   Google Scholar [9] Yu. Chekanov and F. Schlenk, Twist tori II: Non-displaceability,, in preparation., ().   Google Scholar [10] C.-H. Cho, Holomorphic discs, spin structures, and Floer cohomology of the Clifford torus, Int. Math. Res. Not., 35 (2004), 1803-1843. doi: doi:10.1155/S1073792804132716.  Google Scholar [11] D. Eisenbud, "Commutative Algebra. With a View Toward Algebraic Geometry," Graduate Texts in Mathematics 150, Springer-Verlag, New York, 1995.  Google Scholar [12] Ya. Eliashberg and L. Polterovich, The problem of Lagrangian knots in four-manifolds, in "Geometric Topology (Athens, GA, 1993)," AMS/IP Stud. Adv. Math. 2.1, AMS, (1997), 313-327.  Google Scholar [13] Ya. Eliashberg and L. Polterovich, Symplectic quasi-states on the quadric surface and Lagrangian submanifolds,, , ().   Google Scholar [14] M. Entov and L. Polterovich, Rigid subsets of symplectic manifolds, Compos. Math., 145 (2009), 773-826. doi: doi:10.1112/S0010437X0900400X.  Google Scholar [15] A. Floer, Morse theory for Lagrangian intersections, J. Differential Geom., 28 (1988), 513-547.  Google Scholar [16] K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, "Lagrangian Intersection Floer Theory: Anomaly and Obstruction. Part I," AMS/IP Studies in Advanced Mathematics 46.1, AMS, International Press, Somerville, MA, 2009.  Google Scholar [17] K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, "Lagrangian Intersection Floer Theory: Anomaly and Obstruction. Part II," AMS/IP Studies in Advanced Mathematics 46.2, AMS, International Press, Somerville, MA, 2009.  Google Scholar [18] K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Toric degeneration and non-displaceable Lagrangian tori in $S^2 \times S^2$,, \arXiv{1002.1660}., ().   Google Scholar [19] A. Gadbled, Exotic Hamiltonian tori in $\CP^2$ and $S^2 \times S^2$,, in preparation., ().   Google Scholar [20] M. Gromov, Pseudo-holomorphic curves in symplectic manifolds, Invent. math., 82 (1985), 307-347.  Google Scholar [21] H. Hofer, On the topological properties of symplectic maps, Proc. Roy. Soc. Edinburgh Sect. A , 115 (1990), 25-38.  Google Scholar [22] C. Weibel, "An Introduction to Homological Algebra," Cambridge Studies in Advanced Mathematics 38, Cambridge University Press, Cambridge, 1994.  Google Scholar

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##### References:
 [1] P. Albers and U. Frauenfelder, A non-displaceable Lagrangian torus in $T^$*$S^2$, Comm. Pure Appl. Math., 61 (2008), 1046-1051. doi: doi:10.1002/cpa.20216.  Google Scholar [2] V. I. Arnold, On a characteristic class entering into conditions of quantization, Funkcional. Anal. i Prilozen., 1 (1967), 1-14. doi: doi:10.1007/BF01075861.  Google Scholar [3] D. Auroux, Mirror symmetry and T-duality in the complement of an anticanonical divisor, J. Gökova Geom. Topol., 1 (2007), 51-91.  Google Scholar [4] P. Biran and O. Cornea, A Lagrangian quantum homology, in "New Perspectives and Challenges in Symplectic Field Theory," CRM Proc. Lecture Notes 49, AMS, (2009), 1-44.  Google Scholar [5] P. Biran and O. Cornea, Rigidity and uniruling for Lagrangian submanifolds, Geom. Topol., 13 (2009), 2881-2989. doi: doi:10.2140/gt.2009.13.2881.  Google Scholar [6] Yu. V. Chekanov, Lagrangian tori in a symplectic vector space and global symplectomorphisms, Math. Z., 223 (1996), 547-559.  Google Scholar [7] Yu. V. Chekanov, Lagrangian intersections, symplectic energy, and areas of holomorphic curves, Duke Math. J., 95 (1998), 213-226. doi: doi:10.1215/S0012-7094-98-09506-0.  Google Scholar [8] Yu. Chekanov and F. Schlenk, Twist tori I: Construction and classification,, in preparation., ().   Google Scholar [9] Yu. Chekanov and F. Schlenk, Twist tori II: Non-displaceability,, in preparation., ().   Google Scholar [10] C.-H. Cho, Holomorphic discs, spin structures, and Floer cohomology of the Clifford torus, Int. Math. Res. Not., 35 (2004), 1803-1843. doi: doi:10.1155/S1073792804132716.  Google Scholar [11] D. Eisenbud, "Commutative Algebra. With a View Toward Algebraic Geometry," Graduate Texts in Mathematics 150, Springer-Verlag, New York, 1995.  Google Scholar [12] Ya. Eliashberg and L. Polterovich, The problem of Lagrangian knots in four-manifolds, in "Geometric Topology (Athens, GA, 1993)," AMS/IP Stud. Adv. Math. 2.1, AMS, (1997), 313-327.  Google Scholar [13] Ya. Eliashberg and L. Polterovich, Symplectic quasi-states on the quadric surface and Lagrangian submanifolds,, , ().   Google Scholar [14] M. Entov and L. Polterovich, Rigid subsets of symplectic manifolds, Compos. Math., 145 (2009), 773-826. doi: doi:10.1112/S0010437X0900400X.  Google Scholar [15] A. Floer, Morse theory for Lagrangian intersections, J. Differential Geom., 28 (1988), 513-547.  Google Scholar [16] K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, "Lagrangian Intersection Floer Theory: Anomaly and Obstruction. Part I," AMS/IP Studies in Advanced Mathematics 46.1, AMS, International Press, Somerville, MA, 2009.  Google Scholar [17] K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, "Lagrangian Intersection Floer Theory: Anomaly and Obstruction. Part II," AMS/IP Studies in Advanced Mathematics 46.2, AMS, International Press, Somerville, MA, 2009.  Google Scholar [18] K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Toric degeneration and non-displaceable Lagrangian tori in $S^2 \times S^2$,, \arXiv{1002.1660}., ().   Google Scholar [19] A. Gadbled, Exotic Hamiltonian tori in $\CP^2$ and $S^2 \times S^2$,, in preparation., ().   Google Scholar [20] M. Gromov, Pseudo-holomorphic curves in symplectic manifolds, Invent. math., 82 (1985), 307-347.  Google Scholar [21] H. Hofer, On the topological properties of symplectic maps, Proc. Roy. Soc. Edinburgh Sect. A , 115 (1990), 25-38.  Google Scholar [22] C. Weibel, "An Introduction to Homological Algebra," Cambridge Studies in Advanced Mathematics 38, Cambridge University Press, Cambridge, 1994.  Google Scholar
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