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Bounding Lagrangian widths via geodesic paths

  • Matthew Strom Borman (a1) and Mark McLean (a2)

The width of a Lagrangian is the largest capacity of a ball that can be symplectically embedded into the ambient manifold such that the ball intersects the Lagrangian exactly along the real part of the ball. Due to Dimitroglou Rizell, finite width is an obstruction to a Lagrangian admitting an exact Lagrangian cap in the sense of Eliashberg–Murphy. In this paper we introduce a new method for bounding the width of a Lagrangian $Q$ by considering the Lagrangian Floer cohomology of an auxiliary Lagrangian $L$ with respect to a Hamiltonian whose chords correspond to geodesic paths in $Q$ . This is formalized as a wrapped version of the Floer–Hofer–Wysocki capacity and we establish an associated energy–capacity inequality with the help of a closed–open map. For any orientable Lagrangian $Q$ admitting a metric of non-positive sectional curvature in a Liouville manifold, we show the width of $Q$ is bounded above by four times its displacement energy.

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[AS06a]Abbondandolo, A. and Schwarz, M., Note on Floer homology and loop space homology, in Morse theoretic methods in nonlinear analysis and in symplectic topology, NATO Science Series II Mathematics, Physics and Chemistry, vol. 217 (Springer, Dordrecht, 2006), 75108.
[AS06b]Abbondandolo, A. and Schwarz, M., On the Floer homology of cotangent bundles, Comm. Pure Appl. Math. 59 (2006), 254316.
[AS10]Abbondandolo, A. and Schwarz, M., Floer homology of cotangent bundles and the loop product, Geom. Topol. 14 (2010), 15691722.
[Abo12]Abouzaid, M., On the wrapped Fukaya category and based loops, J. Symplectic Geom. 10 (2012), 2779.
[AboS10]Abouzaid, M. and Seidel, P., An open string analogue of Viterbo functoriality, Geom. Topol. 14 (2010), 627718.
[Alb05]Albers, P., On the extrinsic topology of Lagrangian submanifolds, Int. Math. Res. Not. IMRN 2005 (2005), 23412371.
[Alb08]Albers, P., A Lagrangian Piunikhin–Salamon–Schwarz morphism and two comparison homomorphisms in Floer homology, Int. Math. Res. Not. IMRN, Art. ID rnm 134, 56pp (2008).
[Alb10]Albers, P., Erratum for “On the extrinsic topology of Lagrangian submanifolds”, Int. Math. Res. Not. IMRN 2010 (2010), 13631369.
[Ale71]Alexander, H., Continuing 1-dimensional analytic sets, Math. Ann. 191 (1971), 143144.
[ALP94]Audin, M., Lalonde, F. and Polterovich, L., Symplectic rigidity: Lagrangian submanifolds, in Holomorphic curves in symplectic geometry, Progress in Mathematics, vol. 117 (Birkhäuser, Basel, 1994), 271321.
[BC06]Barraud, J.-F. and Cornea, O., Homotopic dynamics in symplectic topology, in Morse theoretic methods in nonlinear analysis and in symplectic topology, NATO Science Series II: Mathematics, Physics and Chemistry, vol. 217 (Springer, Dordrecht, 2006), 109148.
[BC07]Barraud, J.-F. and Cornea, O., Lagrangian intersections and the Serre spectral sequence, Ann. of Math. (2) 166 (2007), 657722.
[Bir99]Biran, P., A stability property of symplectic packing, Invent. Math. 136 (1999), 123155.
[Bir01]Biran, P., From symplectic packing to algebraic geometry and back, in European Congress of Mathematics, Vol. II (Barcelona, 2000), Progress in Mathematics, vol. 202 (Birkhäuser, Basel, 2001), 507524.
[BC09]Biran, P. and Cornea, O., Rigidity and uniruling for Lagrangian submanifolds, Geom. Topol. 13 (2009), 28812989.
[BPS03]Biran, P., Polterovich, L. and Salamon, D., Propagation in Hamiltonian dynamics and relative symplectic homology, Duke Math. J. 119 (2003), 65118.
[Buh10]Buhovsky, L., A maximal relative symplectic packing construction, J. Symplectic Geom. 8 (2010), 6772.
[BH11]Buse, O. and Hind, R., Symplectic embeddings of ellipsoids in dimension greater than four, Geom. Topol. 15 (2011), 20912110.
[CFHW96]Cieliebak, K., Floer, A., Hofer, H. and Wysocki, K., Applications of symplectic homology. II. Stability of the action spectrum, Math. Z. 223 (1996), 2745.
[CL09]Cieliebak, K. and Latschev, J., The role of string topology in symplectic field theory, in New perspectives and challenges in symplectic field theory, CRM Proceedings Lecture Notes, vol. 49 (American Mathematical Society, Providence, RI, 2009), 113146.
[Cha12a]Chantraine, B., Some non-collarable slices of Lagrangian surfaces, Bull. Lond. Math. Soc. 44 (2012), 981987.
[Cha12b]Charette, F., A geometric refinement of a theorem of Chekanov, J. Symplectic Geom. 10 (2012), 475491.
[Cha14]Charette, F., Uniruling for orientable Lagrangian surfaces, Preprint (2014), arXiv:1401:1953.
[CL05]Cornea, O. and Lalonde, F., Cluster homology, Preprint (2005), arXiv:math.SG/0508345v1.
[CL06]Cornea, O. and Lalonde, F., Cluster homology: an overview of the construction and results, Electron. Res. Announc. Amer. Math. Soc. 12 (2006), 112 (electronic).
[Dam12]Damian, M., Floer homology on the universal cover, Audin’s conjecture and other constraints on Lagrangian submanifolds, Comment. Math. Helv. 87 (2012), 433462.
[DR13]Rizell, G. D., Exact Lagrangian caps and non-uniruled Lagrangian submanifolds, Preprint (2013), arXiv:1306.4667.
[Dra08]Dragnev, D. L., Symplectic rigidity, symplectic fixed points, and global perturbations of Hamiltonian systems, Comm. Pure Appl. Math. 61 (2008), 346370.
[Dui76]Duistermaat, J. J., On the Morse index in variational calculus, Adv. Math. 21 (1976), 173195.
[EEMS13]Ekholm, T., Eliashberg, Y., Murphy, E. and Smith, I., Constructing exact Lagrangian immersions with few double points, Geom. Funct. Anal. 23 (2013), 17721803.
[Eli91]Eliashberg, Y., New invariants of open symplectic and contact manifolds, J. Amer. Math. Soc. 4 (1991), 513520.
[EM13]Eliashberg, Y. and Murphy, E., Lagrangian caps, Geom. Funct. Anal. 23 (2013), 14831514.
[EK11]Evans, J. D. and Kędra, J., Remarks on monotone Lagrangians in $\mathbb{C}^{n}$, Preprint (2011),arXiv:1110.0927.
[FP82]Fefferman, C. and Phong, D. H., Symplectic geometry and positivity of pseudodifferential operators, Proc. Natl Acad. Sci. USA 79 (1982), 710713.
[Fis11]Fish, J. W., Target-local Gromov compactness, Geom. Topol. 2 (2011), 765826.
[Flo88a]Floer, A., Morse theory for Lagrangian intersections, J. Differential Geom. 28 (1988), 513547.
[Flo88b]Floer, A., The unregularized gradient flow of the symplectic action, Comm. Pure Appl. Math. 41 (1988), 775813.
[Flo89a]Floer, A., Symplectic fixed points and holomorphic spheres, Comm. Math. Phys. 120 (1989), 575611.
[Flo89b]Floer, A., Witten’s complex and infinite-dimensional Morse theory, J. Differential Geom. 30 (1989), 207221.
[FH94]Floer, A. and Hofer, H., Symplectic homology. I. Open sets in Cn, Math. Z. 215 (1994), 3788.
[FHS95]Floer, A., Hofer, H. and Salamon, D., Transversality in elliptic Morse theory for the symplectic action, Duke Math. J. 80 (1995), 251292.
[FHW94]Floer, A., Hofer, H. and Wysocki, K., Applications of symplectic homology. I, Math. Z. 217 (1994), 577606.
[Fuk06]Fukaya, K., Application of Floer homology of Langrangian submanifolds to symplectic topology, in Morse theoretic methods in nonlinear analysis and in symplectic topology, NATO Science Series II: Mathematics, Physics and Chemistry, vol. 217 (Springer, Dordrecht, 2006), 231276.
[Gin07]Ginzburg, V. L., Coisotropic intersections, Duke Math. J. 140 (2007), 111163.
[Gin10]Ginzburg, V. L., The Conley conjecture, Ann. of Math. (2) 172 (2010), 11271180.
[Gin11]Ginzburg, V. L., On Maslov class rigidity for coisotropic submanifolds, Pacific J. Math. 250 (2011), 139161.
[GR84]Grauert, H. and Remmert, R., Coherent analytic sheaves, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 265 (Springer-Verlag, Berlin, 1984).
[Gro71]Gromov, M. L., A topological technique for the construction of solutions of differential equations and inequalities, Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2 (Gauthier-Villars, Paris, 1971), 221225.
[Gro85]Gromov, M., Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307347.
[Gut10]Guth, L., Metaphors in systolic geometry, in Proceedings of the international congress of mathematicians. Vol. II (Hindustan Book Agency, New Delhi, 2010), 745768.
[Her00]Hermann, D., Holomorphic curves and Hamiltonian systems in an open set with restricted contact-type boundary, Duke Math. J. 103 (2000), 335374.
[Her04]Hermann, D., Inner and outer Hamiltonian capacities, Bull. Soc. Math. France 132 (2004), 509541.
[HK14]Hind, R. and Kerman, E., New obstructions to symplectic embeddings, Invent. Math. 196 (2014), 383452.
[Hof90]Hofer, H., On the topological properties of symplectic maps, Proc. Roy. Soc. Edinburgh Sect. A 115 (1990), 2538.
[HZ94]Hofer, H. and Zehnder, E., Symplectic invariants and Hamiltonian dynamics, Birkhäuser Advanced Texts: Basler Lehrbücher (Birkhäuser, Basel, 1994).
[HLS13]Humilière, V., Leclercq, R. and Seyfaddini, S., Coisotropic rigidity and $C^{0}$-symplectic geometry, Preprint (2013), arXiv:1305.1287v1.
[Hut10]Hutchings, M., Embedded contact homology and its applications, in Proceedings of the international congress of mathematicians. Vol. II (Hindustan Book Agency, New Delhi, 2010), 10221041.
[Iri12]Irie, K., Symplectic homology of disc cotangent bundles of domains in Euclidean space, Preprint (2012), arXiv:1211.2184.
[Ker09]Kerman, E., Action selectors and Maslov class rigidity, Int. Math. Res. Not. IMRN 23 (2009), 43954427.
[KŞ10]Kerman, E. and Şirikçi, N. I., Maslov class rigidity for Lagrangian submanifolds via Hofer’s geometry, Comment. Math. Helv. 85 (2010), 907949.
[Lec08]Leclercq, R., Spectral invariants in Lagrangian Floer theory, J. Mod. Dyn. 2 (2008), 249286.
[Lee76]Lees, J. A., On the classification of Lagrange immersions, Duke Math. J. 43 (1976), 217224.
[LR13]Lisi, S. and Rieser, A., Coisotropic Hofer–Zehnder capacities and non-squeezing for relative embeddings, Preprint (2013), arXiv:1312.7334.
[MP94]McDuff, D. and Polterovich, L., Symplectic packings and algebraic geometry, Invent. Math. 115 (1994), 405434. With appendix by Y. Karshon.
[MS12]McDuff, D. and Schlenk, F., The embedding capacity of 4-dimensional symplectic ellipsoids, Ann. of Math. (2) 175 (2012), 11911282.
[Mil63]Milnor, J., Morse theory: based on lecture notes by M. Spivak and R. Wells, Annals of Mathematics Studies, vol. 51 (Princeton University Press, Princeton, NJ, 1963).
[Mur12]Murphy, E., Loose Legendrian embeddings in high dimensional contact manifolds, Preprint (2012), arXiv:1201.2245.
[Oh93]Oh, Y.-G., Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. I, Comm. Pure Appl. Math. 46 (1993), 949993.
[Oh95]Oh, Y.-G., Addendum to: “Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. I”, Comm. Pure Appl. Math. 48 (1995), 12991302.
[PPS03]Paternain, G. P., Polterovich, L. and Siburg, K. F., Boundary rigidity for Lagrangian submanifolds, non-removable intersections, and Aubry–Mather theory, Mosc. Math. J. 3 (2003), 593619; 745. Dedicated to Vladimir I. Arnold on the occasion of his 65th birthday.
[Pol01]Polterovich, L., The geometry of the group of symplectic diffeomorphisms, Lectures in Mathematics ETH Zürich (Birkhäuser, Basel, 2001).
[Rie10]Rieser, A., Lagrangian blow-ups, blow-downs, and applications to real packing, Preprint (2010), arXiv:1012.1034v2.
[RS93]Robbin, J. and Salamon, D., The Maslov index for paths, Topology 32 (1993), 827844.
[RS95]Robbin, J. and Salamon, D., The spectral flow and the Maslov index, Bull. Lond. Math. Soc. 27 (1995), 133.
[SW06]Salamon, D. A. and Weber, J., Floer homology and the heat flow, Geom. Funct. Anal. 16 (2006), 10501138.
[SZ92]Salamon, D. and Zehnder, E., Morse theory for periodic solutions of Hamiltonian systems and the Maslov index, Comm. Pure Appl. Math. 45 (1992), 13031360.
[Sch05a]Schlenk, F., Embedding problems in symplectic geometry, de Gruyter Expositions in Mathematics, vol. 40 (Walter de Gruyter, Berlin, 2005).
[Sch05b]Schlenk, F., Packing symplectic manifolds by hand, J. Symplectic Geom. 3 (2005), 313340.
[Sik89]Sikorav, J.-C., Rigidité symplectique dans le cotangent de T n, Duke Math. J. 59 (1989), 759763.
[Sik91]Sikorav, J.-C., Quelques propriétés des plongements Lagrangiens, in Mém. Soc. Math. France (N.S.) (1991), 151167. Analyse globale et physique mathématique (Lyon, 1989).
[Sik94]Sikorav, J.-C., Some properties of holomorphic curves in almost complex manifolds, in Holomorphic curves in symplectic geometry, Progress in Mathematics, vol. 117 (Birkhäuser, Basel, 1994), 165189.
[Tra95]Traynor, L., Symplectic packing constructions, J. Differential Geom. 42 (1995), 411429.
[Vit90a]Viterbo, C., A new obstruction to embedding Lagrangian tori, Invent. Math. 100 (1990), 301320.
[Vit90b]Viterbo, C., Plongements lagrangiens et capacités symplectiques de tores dans R2n, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), 487490.
[Vit99]Viterbo, C., Functors and computations in Floer homology with applications. I, Geom. Funct. Anal. 9 (1999), 9851033.
[Wei71]Weinstein, A., Symplectic manifolds and their Lagrangian submanifolds, Adv. Math. 6 (1971), 329356.
[Zeh13]Zehmisch, K., The codisc radius capacity, Electron. Res. Announc. Amer. Math. Soc. 20 (2013), 7796 (electronic).
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