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. doi: doi:10.1002/cpa.20216.

[2]

V. I. Arnold, On a characteristic class entering into conditions of quantization,, Funkcional. Anal. i Prilozen., 1 (1967), 1. doi: doi:10.1007/BF01075861.

[3]

D. Auroux, Mirror symmetry and T-duality in the complement of an anticanonical divisor,, J. Gökova Geom. Topol., 1 (2007), 51.

[4]

P. Biran and O. Cornea, A Lagrangian quantum homology,, in, 49 (2009), 1.

[5]

P. Biran and O. Cornea, Rigidity and uniruling for Lagrangian submanifolds,, Geom. Topol., 13 (2009), 2881. doi: doi:10.2140/gt.2009.13.2881.

[6]

Yu. V. Chekanov, Lagrangian tori in a symplectic vector space and global symplectomorphisms,, Math. Z., 223 (1996), 547.

[7]

Yu. V. Chekanov, Lagrangian intersections, symplectic energy, and areas of holomorphic curves,, Duke Math. J., 95 (1998), 213. doi: doi:10.1215/S0012-7094-98-09506-0.

[8]

Yu. Chekanov and F. Schlenk, Twist tori I: Construction and classification,, in preparation., ().

[9]

Yu. Chekanov and F. Schlenk, Twist tori II: Non-displaceability,, in preparation., ().

[10]

C.-H. Cho, Holomorphic discs, spin structures, and Floer cohomology of the Clifford torus,, Int. Math. Res. Not., 35 (2004), 1803. doi: doi:10.1155/S1073792804132716.

[11]

D. Eisenbud, "Commutative Algebra. With a View Toward Algebraic Geometry,", Graduate Texts in Mathematics 150, 150 (1995).

[12]

Ya. Eliashberg and L. Polterovich, The problem of Lagrangian knots in four-manifolds,, in, 2.1 (1997), 313.

[13]

Ya. Eliashberg and L. Polterovich, Symplectic quasi-states on the quadric surface and Lagrangian submanifolds,, , ().

[14]

M. Entov and L. Polterovich, Rigid subsets of symplectic manifolds,, Compos. Math., 145 (2009), 773. doi: doi:10.1112/S0010437X0900400X.

[15]

A. Floer, Morse theory for Lagrangian intersections,, J. Differential Geom., 28 (1988), 513.

[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, 46.1 (2009).

[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, 46.2 (2009).

[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}., ().

[19]

A. Gadbled, Exotic Hamiltonian tori in $\CP^2$ and $S^2 \times S^2$,, in preparation., ().

[20]

M. Gromov, Pseudo-holomorphic curves in symplectic manifolds,, Invent. math., 82 (1985), 307.

[21]

H. Hofer, On the topological properties of symplectic maps,, Proc. Roy. Soc. Edinburgh Sect. A, 115 (1990), 25.

[22]

C. Weibel, "An Introduction to Homological Algebra,", Cambridge Studies in Advanced Mathematics 38, 38 (1994).

show all references

References:
[1]

P. Albers and U. Frauenfelder, A non-displaceable Lagrangian torus in $T^$*$S^2$,, Comm. Pure Appl. Math., 61 (2008), 1046. doi: doi:10.1002/cpa.20216.

[2]

V. I. Arnold, On a characteristic class entering into conditions of quantization,, Funkcional. Anal. i Prilozen., 1 (1967), 1. doi: doi:10.1007/BF01075861.

[3]

D. Auroux, Mirror symmetry and T-duality in the complement of an anticanonical divisor,, J. Gökova Geom. Topol., 1 (2007), 51.

[4]

P. Biran and O. Cornea, A Lagrangian quantum homology,, in, 49 (2009), 1.

[5]

P. Biran and O. Cornea, Rigidity and uniruling for Lagrangian submanifolds,, Geom. Topol., 13 (2009), 2881. doi: doi:10.2140/gt.2009.13.2881.

[6]

Yu. V. Chekanov, Lagrangian tori in a symplectic vector space and global symplectomorphisms,, Math. Z., 223 (1996), 547.

[7]

Yu. V. Chekanov, Lagrangian intersections, symplectic energy, and areas of holomorphic curves,, Duke Math. J., 95 (1998), 213. doi: doi:10.1215/S0012-7094-98-09506-0.

[8]

Yu. Chekanov and F. Schlenk, Twist tori I: Construction and classification,, in preparation., ().

[9]

Yu. Chekanov and F. Schlenk, Twist tori II: Non-displaceability,, in preparation., ().

[10]

C.-H. Cho, Holomorphic discs, spin structures, and Floer cohomology of the Clifford torus,, Int. Math. Res. Not., 35 (2004), 1803. doi: doi:10.1155/S1073792804132716.

[11]

D. Eisenbud, "Commutative Algebra. With a View Toward Algebraic Geometry,", Graduate Texts in Mathematics 150, 150 (1995).

[12]

Ya. Eliashberg and L. Polterovich, The problem of Lagrangian knots in four-manifolds,, in, 2.1 (1997), 313.

[13]

Ya. Eliashberg and L. Polterovich, Symplectic quasi-states on the quadric surface and Lagrangian submanifolds,, , ().

[14]

M. Entov and L. Polterovich, Rigid subsets of symplectic manifolds,, Compos. Math., 145 (2009), 773. doi: doi:10.1112/S0010437X0900400X.

[15]

A. Floer, Morse theory for Lagrangian intersections,, J. Differential Geom., 28 (1988), 513.

[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, 46.1 (2009).

[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, 46.2 (2009).

[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}., ().

[19]

A. Gadbled, Exotic Hamiltonian tori in $\CP^2$ and $S^2 \times S^2$,, in preparation., ().

[20]

M. Gromov, Pseudo-holomorphic curves in symplectic manifolds,, Invent. math., 82 (1985), 307.

[21]

H. Hofer, On the topological properties of symplectic maps,, Proc. Roy. Soc. Edinburgh Sect. A, 115 (1990), 25.

[22]

C. Weibel, "An Introduction to Homological Algebra,", Cambridge Studies in Advanced Mathematics 38, 38 (1994).

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