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  2013, 20: 77-96. doi: 10.3934/era.2013.20.77

The codisc radius capacity

1. 

Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Köln, Germany

Received  December 2012 Revised  June 2013 Published  October 2013

We prove a generalization of Gromov's packing inequality to symplectic embeddings of the boundaries of two balls such that the bounded components of the complements of the image spheres are disjoint. Moreover, we define a capacity which measures the size of Weinstein tubular neighborhoods of Lagrangian submanifolds. In symplectic vector spaces this leads to bounds on the codisc radius for any closed Lagrangian submanifold in terms of Viterbo's isoperimetric inequality. Furthermore, we introduce the spherical variant of the relative Gromov radius and prove its finiteness for monotone Lagrangian tori in symplectic vector spaces.
Citation: Kai Zehmisch. The codisc radius capacity. Electronic Research Announcements, 2013, 20: 77-96. doi: 10.3934/era.2013.20.77
References:
[1]

C. Abbas, Finite energy surfaces and the chord problem, Duke Math. J., 96 (1999), 241-316. doi: 10.1215/S0012-7094-99-09608-4.

[2]

C. Abbas, Introduction to Compactness Results in Symplectic Field Theory, to appear, Springer, 2013.

[3]

V. I. Arnol'd, The first steps of symplectic topology, Uspekhi Mat. Nauk, 41 (1986), 3-18, 229.

[4]

S. Artstein-Avidan and Y. Ostrover, A Brunn-Minkowski inequality for symplectic capacities of convex domains, Int. Math. Res. Not. IMRN, (2008), Art. ID rnn044, 31 pp. doi: 10.1093/imrn/rnn044.

[5]

J.-F. Barraud and O. Cornea, Homotopic dynamics in symplectic topology, in Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology, NATO Sci. Ser. II Math. Phys. Chem., 217, Springer, Dordrecht, 2006, 109-148. doi: 10.1007/1-4020-4266-3_03.

[6]

J.-F. Barraud and O. Cornea, Lagrangian intersections and the Serre spectral sequence, Ann. of Math. (2), 166 (2007), 657-722. doi: 10.4007/annals.2007.166.657.

[7]

S. M. Bates, A capacity representation theorem for some non-convex domains, Math. Z., 227 (1998), 571-581. doi: 10.1007/PL00004394.

[8]

P. Biran, Symplectic packing in dimension $4$, Geom. Funct. Anal., 7 (1997), 420-437. doi: 10.1007/s000390050014.

[9]

P. Biran and O. Cornea, A Lagrangian quantum homology, in New Perspectives and Challenges in Symplectic Field Theory, CRM Proc. Lecture Notes, 49, Amer. Math. Soc., Providence, RI, 2009, 1-44.

[10]

P. Biran and O. Cornea, Rigidity and uniruling for Lagrangian submanifolds, Geom. Topol., 13 (2009), 2881-2989. doi: 10.2140/gt.2009.13.2881.

[11]

F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki and E. Zehnder, Compactness results in symplectic field theory, Geom. Topol., 7 (2003), 799-888. doi: 10.2140/gt.2003.7.799.

[12]

M. Brown, A proof of the generalized Schoenflies theorem, Bull. Amer. Math. Soc., 66 (1960), 74-76. doi: 10.1090/S0002-9904-1960-10400-4.

[13]

L. Buhovsky, The Maslov class of Lagrangian tori and quantum products in Floer cohomology, J. Topol. Anal., 2 (2010), 57-75. doi: 10.1142/S1793525310000240.

[14]

Yu. V. Chekanov, Lagrangian intersections, symplectic energy, and areas of holomorphic curves, Duke Math. J., 95 (1998), 213-226. doi: 10.1215/S0012-7094-98-09506-0.

[15]

K. Cieliebak and Y. Eliashberg, From Stein to Weinstein and Back: Symplectic Geometry of Affine Complex Manifolds, Amer. Math. Soc. Colloq. Publ., 59, American Mathematical Society, Providence, RI, 2012.

[16]

K. Cieliebak, H. Hofer, J. Latschev and F. Schlenk, Quantitative symplectic geometry, in Dynamics, Ergodic Theory, and Geometry, Math. Sci. Res. Inst. Publ., 54, Cambridge Univ. Press, Cambridge, 2007, 1-44. doi: 10.1017/CBO9780511755187.002.

[17]

K. Cieliebak and K. Mohnke, Punctured holomorphic curves and Lagrangian embeddings,, in preparation., (). 

[18]

M. Damian, Floer homology on the universal cover, Audin's conjecture and other constraints on Lagrangian submanifolds, Comment. Math. Helv., 87 (2012), 433-462. doi: 10.4171/CMH/259.

[19]

I. Ekeland and H. Hofer, Symplectic topology and Hamiltonian dynamics. II, Math. Z., 203 (1990), 553-567.

[20]

Y. Eliashberg and L. Polterovich, Local Lagrangian $2$-knots are trivial, Ann. of Math. (2), 144 (1996), 61-76. doi: 10.2307/2118583.

[21]

U. Frauenfelder, Gromov convergence of pseudoholomorphic disks, J. Fixed Point Theory Appl., 3 (2008), 215-271. doi: 10.1007/s11784-008-0078-1.

[22]

H. Geiges, An Introduction to Contact Topology, Cambridge Stud. Adv. Math., 109, Cambridge University Press, Cambridge, 2008. doi: 10.1017/CBO9780511611438.

[23]

H. Geiges and K. Zehmisch, Eliashberg's proof of Cerf's theorem, J. Topol. Anal., 2 (2010), 543-579. doi: 10.1142/S1793525310000446.

[24]

H. Geiges and K. Zehmisch, Symplectic cobordisms and the strong Weinstein conjecture, Math. Proc. Cambridge Philos. Soc., 153 (2012), 261-279. doi: 10.1017/S0305004112000163.

[25]

H. Geiges and K. Zehmisch, How to recognise a 4-ball when you see one, to appear, Münster J. Math., (2013).

[26]

M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math., 82 (1985), 307-347. doi: 10.1007/BF01388806.

[27]

D. Hermann, Inner and outer Hamiltonian capacities, Bull. Soc. Math. France, 132 (2004), 509-541.

[28]

H. Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three, Invent. Math., 114 (1993), 515-563. doi: 10.1007/BF01232679.

[29]

H. Hofer, V. Lizan and J.-C. Sikorav, On genericity for holomorphic curves in four-dimensional almost-complex manifolds, J. Geom. Anal., 7 (1997), 149-159. doi: 10.1007/BF02921708.

[30]

H. Hofer, K. Wysocki and E. Zehnder, The dynamics on three-dimensional strictly convex energy surfaces, Ann. of Math. (2), 148 (1998), 197-289. doi: 10.2307/120994.

[31]

H. Hofer, K. Wysocki and E. Zehnder, Finite energy foliations of tight three-spheres and Hamiltonian dynamics, Ann. of Math. (2), 157 (2003), 125-255. doi: 10.4007/annals.2003.157.125.

[32]

H. Hofer and E. Zehnder, A new capacity for symplectic manifolds, in Analysis, et cetera, Academic Press, Boston, MA, 1990, 405-427.

[33]

H. Hofer and E. Zehnder, Symplectic invariants and Hamiltonian dynamics, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 1994. doi: 10.1007/978-3-0348-8540-9.

[34]

M.-Y. Jiang, An inequality for symplectic capacity, Bull. London Math. Soc., 31 (1999), 237-240. doi: 10.1112/S0024609398004962.

[35]

L. Lazzarini, Existence of a somewhere injective pseudo-holomorphic disc, Geom. Funct. Anal., 10 (2000), 829-862. doi: 10.1007/PL00001640.

[36]

B. Mazur, On embeddings of spheres, Bull. Amer. Math. Soc., 65 (1959), 59-65. doi: 10.1090/S0002-9904-1959-10274-3.

[37]

D. McDuff, Symplectic embeddings and continued fractions: A survey, Jpn. J. Math., 4 (2009), 121-139. doi: 10.1007/s11537-009-0926-9.

[38]

D. McDuff and L. Polterovich, Symplectic packings and algebraic geometry, With an appendix by Yael Karshon, Invent. Math., 115 (1994), 405-434. doi: 10.1007/BF01231766.

[39]

D. McDuff and D. Salamon, Introduction to Symplectic Topology, 2nd edition, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1998.

[40]

D. McDuff and D. Salamon, $J$-Holomorphic Curves and Symplectic Topology, Amer. Math. Soc. Colloq. Publ., 52, American Mathematical Society, Providence, RI, 2004.

[41]

D. McDuff and F. Schlenk, The embedding capacity of 4-dimensional symplectic ellipsoids, Ann. of Math. (2), 175 (2012), 1191-1282. doi: 10.4007/annals.2012.175.3.5.

[42]

M. Morse, A reduction of the Schoenflies extension problem, Bull. Amer. Math. Soc., 66 (1960), 113-115. doi: 10.1090/S0002-9904-1960-10420-X.

[43]

L. Polterovich, The Geometry of the Group of Symplectic Diffeomorphisms, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2001. doi: 10.1007/978-3-0348-8299-6.

[44]

F. Schlenk, Embedding Problems in Symplectic Geometry, de Gruyter Expositions in Mathematics, 40, Walter de Gruyter GmbH & Co. KG, Berlin, 2005. doi: 10.1515/9783110199697.

[45]

J. Swoboda and F. Ziltener, Coisotropic displacement and small subsets of a symplectic manifold, Math. Z., 271 (2012), 415-445. doi: 10.1007/s00209-011-0870-2.

[46]

J. Swoboda and F. Ziltener}, A symplectically non-squeezable small set and the regular coisotropic capacity,, preprint, (). 

[47]

C. Viterbo, Metric and isoperimetric problems in symplectic geometry, J. Amer. Math. Soc., 13 (2000), 411-431. doi: 10.1090/S0894-0347-00-00328-3.

[48]

A. Weinstein, Symplectic manifolds and their Lagrangian submanifolds, Advances in Math., 6 (1971), 329-346. doi: 10.1016/0001-8708(71)90020-X.

[49]

K. Zehmisch, Singularities and Self-Intersections of Holomorphic Discs, Doktorarbeit, Universität Leipzig, 2008.

[50]

K. Zehmisch, Lagrangian non-squeezing and a geometric inequality,, preprint, (). 

[51]

K. Zehmisch and F. Ziltener, Discontinuous capacities,, preprint, (). 

show all references

References:
[1]

C. Abbas, Finite energy surfaces and the chord problem, Duke Math. J., 96 (1999), 241-316. doi: 10.1215/S0012-7094-99-09608-4.

[2]

C. Abbas, Introduction to Compactness Results in Symplectic Field Theory, to appear, Springer, 2013.

[3]

V. I. Arnol'd, The first steps of symplectic topology, Uspekhi Mat. Nauk, 41 (1986), 3-18, 229.

[4]

S. Artstein-Avidan and Y. Ostrover, A Brunn-Minkowski inequality for symplectic capacities of convex domains, Int. Math. Res. Not. IMRN, (2008), Art. ID rnn044, 31 pp. doi: 10.1093/imrn/rnn044.

[5]

J.-F. Barraud and O. Cornea, Homotopic dynamics in symplectic topology, in Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology, NATO Sci. Ser. II Math. Phys. Chem., 217, Springer, Dordrecht, 2006, 109-148. doi: 10.1007/1-4020-4266-3_03.

[6]

J.-F. Barraud and O. Cornea, Lagrangian intersections and the Serre spectral sequence, Ann. of Math. (2), 166 (2007), 657-722. doi: 10.4007/annals.2007.166.657.

[7]

S. M. Bates, A capacity representation theorem for some non-convex domains, Math. Z., 227 (1998), 571-581. doi: 10.1007/PL00004394.

[8]

P. Biran, Symplectic packing in dimension $4$, Geom. Funct. Anal., 7 (1997), 420-437. doi: 10.1007/s000390050014.

[9]

P. Biran and O. Cornea, A Lagrangian quantum homology, in New Perspectives and Challenges in Symplectic Field Theory, CRM Proc. Lecture Notes, 49, Amer. Math. Soc., Providence, RI, 2009, 1-44.

[10]

P. Biran and O. Cornea, Rigidity and uniruling for Lagrangian submanifolds, Geom. Topol., 13 (2009), 2881-2989. doi: 10.2140/gt.2009.13.2881.

[11]

F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki and E. Zehnder, Compactness results in symplectic field theory, Geom. Topol., 7 (2003), 799-888. doi: 10.2140/gt.2003.7.799.

[12]

M. Brown, A proof of the generalized Schoenflies theorem, Bull. Amer. Math. Soc., 66 (1960), 74-76. doi: 10.1090/S0002-9904-1960-10400-4.

[13]

L. Buhovsky, The Maslov class of Lagrangian tori and quantum products in Floer cohomology, J. Topol. Anal., 2 (2010), 57-75. doi: 10.1142/S1793525310000240.

[14]

Yu. V. Chekanov, Lagrangian intersections, symplectic energy, and areas of holomorphic curves, Duke Math. J., 95 (1998), 213-226. doi: 10.1215/S0012-7094-98-09506-0.

[15]

K. Cieliebak and Y. Eliashberg, From Stein to Weinstein and Back: Symplectic Geometry of Affine Complex Manifolds, Amer. Math. Soc. Colloq. Publ., 59, American Mathematical Society, Providence, RI, 2012.

[16]

K. Cieliebak, H. Hofer, J. Latschev and F. Schlenk, Quantitative symplectic geometry, in Dynamics, Ergodic Theory, and Geometry, Math. Sci. Res. Inst. Publ., 54, Cambridge Univ. Press, Cambridge, 2007, 1-44. doi: 10.1017/CBO9780511755187.002.

[17]

K. Cieliebak and K. Mohnke, Punctured holomorphic curves and Lagrangian embeddings,, in preparation., (). 

[18]

M. Damian, Floer homology on the universal cover, Audin's conjecture and other constraints on Lagrangian submanifolds, Comment. Math. Helv., 87 (2012), 433-462. doi: 10.4171/CMH/259.

[19]

I. Ekeland and H. Hofer, Symplectic topology and Hamiltonian dynamics. II, Math. Z., 203 (1990), 553-567.

[20]

Y. Eliashberg and L. Polterovich, Local Lagrangian $2$-knots are trivial, Ann. of Math. (2), 144 (1996), 61-76. doi: 10.2307/2118583.

[21]

U. Frauenfelder, Gromov convergence of pseudoholomorphic disks, J. Fixed Point Theory Appl., 3 (2008), 215-271. doi: 10.1007/s11784-008-0078-1.

[22]

H. Geiges, An Introduction to Contact Topology, Cambridge Stud. Adv. Math., 109, Cambridge University Press, Cambridge, 2008. doi: 10.1017/CBO9780511611438.

[23]

H. Geiges and K. Zehmisch, Eliashberg's proof of Cerf's theorem, J. Topol. Anal., 2 (2010), 543-579. doi: 10.1142/S1793525310000446.

[24]

H. Geiges and K. Zehmisch, Symplectic cobordisms and the strong Weinstein conjecture, Math. Proc. Cambridge Philos. Soc., 153 (2012), 261-279. doi: 10.1017/S0305004112000163.

[25]

H. Geiges and K. Zehmisch, How to recognise a 4-ball when you see one, to appear, Münster J. Math., (2013).

[26]

M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math., 82 (1985), 307-347. doi: 10.1007/BF01388806.

[27]

D. Hermann, Inner and outer Hamiltonian capacities, Bull. Soc. Math. France, 132 (2004), 509-541.

[28]

H. Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three, Invent. Math., 114 (1993), 515-563. doi: 10.1007/BF01232679.

[29]

H. Hofer, V. Lizan and J.-C. Sikorav, On genericity for holomorphic curves in four-dimensional almost-complex manifolds, J. Geom. Anal., 7 (1997), 149-159. doi: 10.1007/BF02921708.

[30]

H. Hofer, K. Wysocki and E. Zehnder, The dynamics on three-dimensional strictly convex energy surfaces, Ann. of Math. (2), 148 (1998), 197-289. doi: 10.2307/120994.

[31]

H. Hofer, K. Wysocki and E. Zehnder, Finite energy foliations of tight three-spheres and Hamiltonian dynamics, Ann. of Math. (2), 157 (2003), 125-255. doi: 10.4007/annals.2003.157.125.

[32]

H. Hofer and E. Zehnder, A new capacity for symplectic manifolds, in Analysis, et cetera, Academic Press, Boston, MA, 1990, 405-427.

[33]

H. Hofer and E. Zehnder, Symplectic invariants and Hamiltonian dynamics, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 1994. doi: 10.1007/978-3-0348-8540-9.

[34]

M.-Y. Jiang, An inequality for symplectic capacity, Bull. London Math. Soc., 31 (1999), 237-240. doi: 10.1112/S0024609398004962.

[35]

L. Lazzarini, Existence of a somewhere injective pseudo-holomorphic disc, Geom. Funct. Anal., 10 (2000), 829-862. doi: 10.1007/PL00001640.

[36]

B. Mazur, On embeddings of spheres, Bull. Amer. Math. Soc., 65 (1959), 59-65. doi: 10.1090/S0002-9904-1959-10274-3.

[37]

D. McDuff, Symplectic embeddings and continued fractions: A survey, Jpn. J. Math., 4 (2009), 121-139. doi: 10.1007/s11537-009-0926-9.

[38]

D. McDuff and L. Polterovich, Symplectic packings and algebraic geometry, With an appendix by Yael Karshon, Invent. Math., 115 (1994), 405-434. doi: 10.1007/BF01231766.

[39]

D. McDuff and D. Salamon, Introduction to Symplectic Topology, 2nd edition, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1998.

[40]

D. McDuff and D. Salamon, $J$-Holomorphic Curves and Symplectic Topology, Amer. Math. Soc. Colloq. Publ., 52, American Mathematical Society, Providence, RI, 2004.

[41]

D. McDuff and F. Schlenk, The embedding capacity of 4-dimensional symplectic ellipsoids, Ann. of Math. (2), 175 (2012), 1191-1282. doi: 10.4007/annals.2012.175.3.5.

[42]

M. Morse, A reduction of the Schoenflies extension problem, Bull. Amer. Math. Soc., 66 (1960), 113-115. doi: 10.1090/S0002-9904-1960-10420-X.

[43]

L. Polterovich, The Geometry of the Group of Symplectic Diffeomorphisms, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2001. doi: 10.1007/978-3-0348-8299-6.

[44]

F. Schlenk, Embedding Problems in Symplectic Geometry, de Gruyter Expositions in Mathematics, 40, Walter de Gruyter GmbH & Co. KG, Berlin, 2005. doi: 10.1515/9783110199697.

[45]

J. Swoboda and F. Ziltener, Coisotropic displacement and small subsets of a symplectic manifold, Math. Z., 271 (2012), 415-445. doi: 10.1007/s00209-011-0870-2.

[46]

J. Swoboda and F. Ziltener}, A symplectically non-squeezable small set and the regular coisotropic capacity,, preprint, (). 

[47]

C. Viterbo, Metric and isoperimetric problems in symplectic geometry, J. Amer. Math. Soc., 13 (2000), 411-431. doi: 10.1090/S0894-0347-00-00328-3.

[48]

A. Weinstein, Symplectic manifolds and their Lagrangian submanifolds, Advances in Math., 6 (1971), 329-346. doi: 10.1016/0001-8708(71)90020-X.

[49]

K. Zehmisch, Singularities and Self-Intersections of Holomorphic Discs, Doktorarbeit, Universität Leipzig, 2008.

[50]

K. Zehmisch, Lagrangian non-squeezing and a geometric inequality,, preprint, (). 

[51]

K. Zehmisch and F. Ziltener, Discontinuous capacities,, preprint, (). 

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