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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. doi: 10.1215/S0012-7094-99-09608-4. Google Scholar

[2]

C. Abbas, Introduction to Compactness Results in Symplectic Field Theory,, to appear, (2013). Google Scholar

[3]

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

[4]

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

[5]

J.-F. Barraud and O. Cornea, Homotopic dynamics in symplectic topology,, in Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology, (2006), 109. doi: 10.1007/1-4020-4266-3_03. Google Scholar

[6]

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

[7]

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

[8]

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

[9]

P. Biran and O. Cornea, A Lagrangian quantum homology,, in New Perspectives and Challenges in Symplectic Field Theory, (2009), 1. Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

K. Cieliebak and Y. Eliashberg, From Stein to Weinstein and Back: Symplectic Geometry of Affine Complex Manifolds,, Amer. Math. Soc. Colloq. Publ., (2012). Google Scholar

[16]

K. Cieliebak, H. Hofer, J. Latschev and F. Schlenk, Quantitative symplectic geometry,, in Dynamics, (2007), 1. doi: 10.1017/CBO9780511755187.002. Google Scholar

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

H. Geiges, An Introduction to Contact Topology,, Cambridge Stud. Adv. Math., 109 (2008). doi: 10.1017/CBO9780511611438. Google Scholar

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[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. doi: 10.1007/BF02921708. Google Scholar

[30]

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

[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. doi: 10.4007/annals.2003.157.125. Google Scholar

[32]

H. Hofer and E. Zehnder, A new capacity for symplectic manifolds,, in Analysis, (1990), 405. Google Scholar

[33]

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

[34]

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

[35]

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

[36]

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

[37]

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

[38]

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

[39]

D. McDuff and D. Salamon, Introduction to Symplectic Topology,, 2nd edition, (1998). Google Scholar

[40]

D. McDuff and D. Salamon, $J$-Holomorphic Curves and Symplectic Topology,, Amer. Math. Soc. Colloq. Publ., (2004). Google Scholar

[41]

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

[42]

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

[43]

L. Polterovich, The Geometry of the Group of Symplectic Diffeomorphisms,, Lectures in Mathematics ETH Zürich, (2001). doi: 10.1007/978-3-0348-8299-6. Google Scholar

[44]

F. Schlenk, Embedding Problems in Symplectic Geometry,, de Gruyter Expositions in Mathematics, (2005). doi: 10.1515/9783110199697. Google Scholar

[45]

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

[46]

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

[47]

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

[48]

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

[49]

K. Zehmisch, Singularities and Self-Intersections of Holomorphic Discs,, Doktorarbeit, (2008). Google Scholar

[50]

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

[51]

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

show all references

References:
[1]

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

[2]

C. Abbas, Introduction to Compactness Results in Symplectic Field Theory,, to appear, (2013). Google Scholar

[3]

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

[4]

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

[5]

J.-F. Barraud and O. Cornea, Homotopic dynamics in symplectic topology,, in Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology, (2006), 109. doi: 10.1007/1-4020-4266-3_03. Google Scholar

[6]

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

[7]

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

[8]

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

[9]

P. Biran and O. Cornea, A Lagrangian quantum homology,, in New Perspectives and Challenges in Symplectic Field Theory, (2009), 1. Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

K. Cieliebak and Y. Eliashberg, From Stein to Weinstein and Back: Symplectic Geometry of Affine Complex Manifolds,, Amer. Math. Soc. Colloq. Publ., (2012). Google Scholar

[16]

K. Cieliebak, H. Hofer, J. Latschev and F. Schlenk, Quantitative symplectic geometry,, in Dynamics, (2007), 1. doi: 10.1017/CBO9780511755187.002. Google Scholar

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

H. Geiges, An Introduction to Contact Topology,, Cambridge Stud. Adv. Math., 109 (2008). doi: 10.1017/CBO9780511611438. Google Scholar

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[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. doi: 10.1007/BF02921708. Google Scholar

[30]

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

[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. doi: 10.4007/annals.2003.157.125. Google Scholar

[32]

H. Hofer and E. Zehnder, A new capacity for symplectic manifolds,, in Analysis, (1990), 405. Google Scholar

[33]

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

[34]

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

[35]

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

[36]

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

[37]

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

[38]

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

[39]

D. McDuff and D. Salamon, Introduction to Symplectic Topology,, 2nd edition, (1998). Google Scholar

[40]

D. McDuff and D. Salamon, $J$-Holomorphic Curves and Symplectic Topology,, Amer. Math. Soc. Colloq. Publ., (2004). Google Scholar

[41]

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

[42]

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

[43]

L. Polterovich, The Geometry of the Group of Symplectic Diffeomorphisms,, Lectures in Mathematics ETH Zürich, (2001). doi: 10.1007/978-3-0348-8299-6. Google Scholar

[44]

F. Schlenk, Embedding Problems in Symplectic Geometry,, de Gruyter Expositions in Mathematics, (2005). doi: 10.1515/9783110199697. Google Scholar

[45]

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

[46]

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

[47]

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

[48]

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

[49]

K. Zehmisch, Singularities and Self-Intersections of Holomorphic Discs,, Doktorarbeit, (2008). Google Scholar

[50]

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

[51]

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

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