American Institute of Mathematical Sciences

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  2021, 17: 353-399. doi: 10.3934/jmd.2021013

Computing the Rabinowitz Floer homology of tentacular hyperboloids

Alexander Fauck 1, , Will J. Merry 2, and  Jagna Wiśniewska 2,

1. 

Department of Mathematics, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany
2. 

Department of Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland

Received  October 09, 2020 Revised  April 19, 2021 Published  September 2021

We compute the Rabinowitz Floer homology for a class of non-compact hyperboloids $ \Sigma\simeq S^{n+k-1}\times\mathbb{R}^{n-k} $. Using an embedding of a compact sphere $ \Sigma_0\simeq S^{2k-1} $ into the hypersurface $ \Sigma $, we construct a chain map from the Floer complex of $ \Sigma $ to the Floer complex of $ \Sigma_0 $. In contrast to the compact case, the Rabinowitz Floer homology groups of $ \Sigma $ are both non-zero and not equal to its singular homology. As a consequence, we deduce that the Weinstein Conjecture holds for any strongly tentacular deformation of such a hyperboloid.

Keywords: Rabinowitz Floer homology, non-compact hypersurfaces, Weinstein conjecture.
Mathematics Subject Classification: Primary: 53D40, 57R17; Secondary: 37J50, 70H05.
Citation: Alexander Fauck, Will J. Merry, Jagna Wiśniewska. Computing the Rabinowitz Floer homology of tentacular hyperboloids. Journal of Modern Dynamics, 2021, 17: 353-399. doi: 10.3934/jmd.2021013
References:
[1]

A. Abbondandolo and W. J. Merry, Floer homology on the time-energy extended phase space, J. Symplectic Geom., 16 (2018), 279-355.  doi: 10.4310/JSG.2018.v16.n2.a1.  Google Scholar

[2]

A. Abbondandolo and M. Schwarz, Estimates and computations in Rabinowitz-Floer homology, J. Topol. Anal., 1 (2009), 307-405.  doi: 10.1142/S1793525309000205.  Google Scholar

[3]

A. Abbondandolo and M. Schwarz, Floer homology of cotangent bundles and the loop product, Geom. Topol., 14 (2010), 1569-1722.  doi: 10.2140/gt.2010.14.1569.  Google Scholar

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P. Albers and U. Frauenfelder, Rabinowitz Floer homology: a survey, Global Differential Geometry, Springer Proc. Math., 17, Springer, Heidelberg, 2012,437–461. doi: 10.1007/978-3-642-22842-1_14.  Google Scholar

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P. Albers, U. Fuchs and W. J. Merry, Orderability and the Weinstein conjecture, Compos. Math., 151 (2015), 2251-2272.  doi: 10.1112/S0010437X15007642.  Google Scholar

[6]

P. Albers, U. Fuchs and W. J. Merry, Positive loops and $l^{\infty }$ contact systolic inequalities, Selecta Math. (N.S.), 23 (2017), 2491-2521.  doi: 10.1007/s00029-017-0338-2.  Google Scholar

[7]

P. Albers and J. Kang, Vanishing of Rabinowitz Floer homology on negative line bundles, Math. Z., 285 (2017), 493-517.  doi: 10.1007/s00209-016-1718-6.  Google Scholar

[8]

P. Albers and W. J. Merry, Orderability, contact non-squeezing, and Rabinowitz Floer homology, J. Symplectic Geom., 16 (2018), 1481-1547.  doi: 10.4310/JSG.2018.v16.n6.a1.  Google Scholar

[9]

M. Audin and M. Damian, Morse Theory and Floer Homology, Universitext, Springer, London; EDP Sciences, Les Ulis, 2014. doi: 10.1007/978-1-4471-5496-9.  Google Scholar

[10]

B. Chantraine, V. Colin and G. D. Rizell, Positive Legendrian isotopies and Floer theory, Ann. Inst. Fourier (Grenoble), 69 (2019), 1679-1737.  doi: 10.5802/aif.3279.  Google Scholar

[11]

K. Cieliebak and U. A. Frauenfelder, A Floer homology for exact contact embeddings, Pacific J. Math., 239 (2009), 251-316.  doi: 10.2140/pjm.2009.239.251.  Google Scholar

[12]

K. Cieliebak and U. Frauenfelder, Morse homology on noncompact manifolds, J. Korean Math. Soc., 48 (2011), 749-774.  doi: 10.4134/JKMS.2011.48.4.749.  Google Scholar

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K. Cieliebak, U. Frauenfelder and A. Oancea, Rabinowitz Floer homology and symplectic homology, Ann. Sci. Éc. Norm. Supér. (4), 43 (2010), 957–1015. doi: 10.24033/asens.2137.  Google Scholar

[14]

K. Cieliebak, U. Frauenfelder and G. P. Paternain, Symplectic topology of Mañé's critical values, Geom. Topol., 14 (2010), 1765-1870.  doi: 10.2140/gt.2010.14.1765.  Google Scholar

[15]

K. Cieliebak, Y. Eliashberg and L. Polterovich, Contact orderability up to conjugation, Regul. Chaotic Dyn., 22 (2017), 585-602.  doi: 10.1134/S1560354717060028.  Google Scholar

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K. Cieliebak and A. Oancea, Symplectic homology and the Eilenberg–Steenrod axioms, Algebr. Geom. Topol., 18 (2018), 1953-2130.  doi: 10.2140/agt.2018.18.1953.  Google Scholar

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A. C. da Silva, Lectures on Symplectic Geometry, Lectures Notes in Mathematics, 1764, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-540-45330-7.  Google Scholar

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A. Fauck, Rabinowitz-Floer homology on Brieskorn spheres, Int. Math. Res. Not. IMRN, 2015 (2015), 5874-5906.  doi: 10.1093/imrn/rnu109.  Google Scholar

[19]

A. Fauck, Rabinowitz-Floer Homology on Brieskorn Manifolds, Ph.D thesis, Humboldt-Universität zu Berlin, 2016. Google Scholar

[20]

A. Floer, Symplectic fixed points and holomorphic spheres, Comm. Math. Phys., 120 (1989), 575-611.  doi: 10.1007/BF01260388.  Google Scholar

[21]

M. Fraser, L. Polterovich and D. Rosen, On Sandon-type metrics for contactomorphism groups, Ann. Math. Qué., 42 (2018), 191–214. doi: 10.1007/s40316-017-0092-z.  Google Scholar

[22]

U. Frauenfelder, The Arnold-Givental conjecture and moment Floer homology, Int. Math. Res. Not. IMRN, 2004 (2004), 2179-2269.  doi: 10.1155/S1073792804133941.  Google Scholar

[23]

S. Ganatra, J. Pardon and V. Shende, Covariantly functorial wrapped Floer theory on Liouville sectors, Publ. Math. Inst. Hautes Études Sci., 131 (2020), 73–200. doi: 10.1007/s10240-019-00112-x.  Google Scholar

[24]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, "Nauka", Moscow, 1989.  Google Scholar

[25]

Y. Groman, Floer theory and reduced cohomology on open manifolds, preprint, arXiv: 1510.04265. Google Scholar

[26] A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002.   Google Scholar
[27]

L. Hörmander, Symplectic classification of quadratic forms, and general Mehler formulas, Math. Z., 219 (1995), 413-449.  doi: 10.1007/BF02572374.  Google Scholar

[28]

F. Laudenbach, Symplectic Geometry and Floer Homology, Soc. Brasil. Mat., Rio de Janeiro, 2004.  Google Scholar

[29]

D. McDuff and D. Salamon, Introduction to Symplectic Topology, 3rd edition, Oxford Graduate Texts in Mathematics, 27, Oxford University Press, 2017. Google Scholar

[30]

W. J. Merry, On the Rabinowitz Floer homology of twisted cotangent bundles, Calc. Var. Partial Differential Equations, 42 (2011), 355-404.  doi: 10.1007/s00526-011-0391-1.  Google Scholar

[31]

E. Miranda and C. Oms, The singular Weinstein conjecture, preprint, arXiv: 2005.09568. Google Scholar

[32]

F. Pasquotto, R. C. Vandervorst and J. Wiśniewska, Rabinowitz Floer homology for tentacular Hamiltonians, Int. Math. Res. Not. IMRN, 6 (2020). doi: 10.1093/imrn/rnaa132.  Google Scholar

[33]

F. Pasquotto and J. Wiśniewska, Bounds for tentacular Hamiltonians, J. Topol. Anal., 12 (2020), 209-265.  doi: 10.1142/S179352531950047X.  Google Scholar

[34]

A. F. Ritter, Topological quantum field theory structure on symplectic cohomology, J. Topol., 6 (2013), 391-489.  doi: 10.1112/jtopol/jts038.  Google Scholar

[35]

J. Robbin and D. Salamon, The Maslov index for paths, Topology, 32 (1993), 827-844.  doi: 10.1016/0040-9383(93)90052-W.  Google Scholar

[36]

S. Suhr and K. Zehmisch, Linking and closed orbits, Abh. Math. Semin. Univ. Hambg., 86 (2016), 133-150.  doi: 10.1007/s12188-016-0118-5.  Google Scholar

[37]

J. B. van den Berg, F. Pasquotto, T. Rot and R. C. A. M. Vandervorst, On periodic orbits in cotangent bundles of non-compact manifolds, J. Symplectic Geom., 14 (2016), 1145-1173.  doi: 10.4310/JSG.2016.v14.n4.a6.  Google Scholar

[38]

J. B. van den Berg, F. Pasquotto and R. C. Vandervorst, Closed characteristics on non-compact hypersurfaces in $\mathbb{R}^{2n}$, Math. Ann., 343 (2009), 247-284.  doi: 10.1007/s00208-008-0271-y.  Google Scholar

[39]

S. Venkatesh, Rabinowitz Floer homology and mirror symmetry, J. Topol., 11 (2018), 144-179.  doi: 10.1112/topo.12050.  Google Scholar

[40]

C. Viterbo, A proof of Weinstein's conjecture in $\mathbb{R}^{2n}$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987), 337-356.  doi: 10.1016/S0294-1449(16)30363-8.  Google Scholar

[41]

J. J. Wiśniewska, Rabinowitz Floer Homology for Tentacular Hamiltonians, Ph.D thesis, Vrije Universiteit Amsterdam, 2017. Google Scholar

show all references

References:
[1]

A. Abbondandolo and W. J. Merry, Floer homology on the time-energy extended phase space, J. Symplectic Geom., 16 (2018), 279-355.  doi: 10.4310/JSG.2018.v16.n2.a1.  Google Scholar

[2]

A. Abbondandolo and M. Schwarz, Estimates and computations in Rabinowitz-Floer homology, J. Topol. Anal., 1 (2009), 307-405.  doi: 10.1142/S1793525309000205.  Google Scholar

[3]

A. Abbondandolo and M. Schwarz, Floer homology of cotangent bundles and the loop product, Geom. Topol., 14 (2010), 1569-1722.  doi: 10.2140/gt.2010.14.1569.  Google Scholar

[4]

P. Albers and U. Frauenfelder, Rabinowitz Floer homology: a survey, Global Differential Geometry, Springer Proc. Math., 17, Springer, Heidelberg, 2012,437–461. doi: 10.1007/978-3-642-22842-1_14.  Google Scholar

[5]

P. Albers, U. Fuchs and W. J. Merry, Orderability and the Weinstein conjecture, Compos. Math., 151 (2015), 2251-2272.  doi: 10.1112/S0010437X15007642.  Google Scholar

[6]

P. Albers, U. Fuchs and W. J. Merry, Positive loops and $l^{\infty }$ contact systolic inequalities, Selecta Math. (N.S.), 23 (2017), 2491-2521.  doi: 10.1007/s00029-017-0338-2.  Google Scholar

[7]

P. Albers and J. Kang, Vanishing of Rabinowitz Floer homology on negative line bundles, Math. Z., 285 (2017), 493-517.  doi: 10.1007/s00209-016-1718-6.  Google Scholar

[8]

P. Albers and W. J. Merry, Orderability, contact non-squeezing, and Rabinowitz Floer homology, J. Symplectic Geom., 16 (2018), 1481-1547.  doi: 10.4310/JSG.2018.v16.n6.a1.  Google Scholar

[9]

M. Audin and M. Damian, Morse Theory and Floer Homology, Universitext, Springer, London; EDP Sciences, Les Ulis, 2014. doi: 10.1007/978-1-4471-5496-9.  Google Scholar

[10]

B. Chantraine, V. Colin and G. D. Rizell, Positive Legendrian isotopies and Floer theory, Ann. Inst. Fourier (Grenoble), 69 (2019), 1679-1737.  doi: 10.5802/aif.3279.  Google Scholar

[11]

K. Cieliebak and U. A. Frauenfelder, A Floer homology for exact contact embeddings, Pacific J. Math., 239 (2009), 251-316.  doi: 10.2140/pjm.2009.239.251.  Google Scholar

[12]

K. Cieliebak and U. Frauenfelder, Morse homology on noncompact manifolds, J. Korean Math. Soc., 48 (2011), 749-774.  doi: 10.4134/JKMS.2011.48.4.749.  Google Scholar

[13]

K. Cieliebak, U. Frauenfelder and A. Oancea, Rabinowitz Floer homology and symplectic homology, Ann. Sci. Éc. Norm. Supér. (4), 43 (2010), 957–1015. doi: 10.24033/asens.2137.  Google Scholar

[14]

K. Cieliebak, U. Frauenfelder and G. P. Paternain, Symplectic topology of Mañé's critical values, Geom. Topol., 14 (2010), 1765-1870.  doi: 10.2140/gt.2010.14.1765.  Google Scholar

[15]

K. Cieliebak, Y. Eliashberg and L. Polterovich, Contact orderability up to conjugation, Regul. Chaotic Dyn., 22 (2017), 585-602.  doi: 10.1134/S1560354717060028.  Google Scholar

[16]

K. Cieliebak and A. Oancea, Symplectic homology and the Eilenberg–Steenrod axioms, Algebr. Geom. Topol., 18 (2018), 1953-2130.  doi: 10.2140/agt.2018.18.1953.  Google Scholar

[17]

A. C. da Silva, Lectures on Symplectic Geometry, Lectures Notes in Mathematics, 1764, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-540-45330-7.  Google Scholar

[18]

A. Fauck, Rabinowitz-Floer homology on Brieskorn spheres, Int. Math. Res. Not. IMRN, 2015 (2015), 5874-5906.  doi: 10.1093/imrn/rnu109.  Google Scholar

[19]

A. Fauck, Rabinowitz-Floer Homology on Brieskorn Manifolds, Ph.D thesis, Humboldt-Universität zu Berlin, 2016. Google Scholar

[20]

A. Floer, Symplectic fixed points and holomorphic spheres, Comm. Math. Phys., 120 (1989), 575-611.  doi: 10.1007/BF01260388.  Google Scholar

[21]

M. Fraser, L. Polterovich and D. Rosen, On Sandon-type metrics for contactomorphism groups, Ann. Math. Qué., 42 (2018), 191–214. doi: 10.1007/s40316-017-0092-z.  Google Scholar

[22]

U. Frauenfelder, The Arnold-Givental conjecture and moment Floer homology, Int. Math. Res. Not. IMRN, 2004 (2004), 2179-2269.  doi: 10.1155/S1073792804133941.  Google Scholar

[23]

S. Ganatra, J. Pardon and V. Shende, Covariantly functorial wrapped Floer theory on Liouville sectors, Publ. Math. Inst. Hautes Études Sci., 131 (2020), 73–200. doi: 10.1007/s10240-019-00112-x.  Google Scholar

[24]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, "Nauka", Moscow, 1989.  Google Scholar

[25]

Y. Groman, Floer theory and reduced cohomology on open manifolds, preprint, arXiv: 1510.04265. Google Scholar

[26] A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002.   Google Scholar
[27]

L. Hörmander, Symplectic classification of quadratic forms, and general Mehler formulas, Math. Z., 219 (1995), 413-449.  doi: 10.1007/BF02572374.  Google Scholar

[28]

F. Laudenbach, Symplectic Geometry and Floer Homology, Soc. Brasil. Mat., Rio de Janeiro, 2004.  Google Scholar

[29]

D. McDuff and D. Salamon, Introduction to Symplectic Topology, 3rd edition, Oxford Graduate Texts in Mathematics, 27, Oxford University Press, 2017. Google Scholar

[30]

W. J. Merry, On the Rabinowitz Floer homology of twisted cotangent bundles, Calc. Var. Partial Differential Equations, 42 (2011), 355-404.  doi: 10.1007/s00526-011-0391-1.  Google Scholar

[31]

E. Miranda and C. Oms, The singular Weinstein conjecture, preprint, arXiv: 2005.09568. Google Scholar

[32]

F. Pasquotto, R. C. Vandervorst and J. Wiśniewska, Rabinowitz Floer homology for tentacular Hamiltonians, Int. Math. Res. Not. IMRN, 6 (2020). doi: 10.1093/imrn/rnaa132.  Google Scholar

[33]

F. Pasquotto and J. Wiśniewska, Bounds for tentacular Hamiltonians, J. Topol. Anal., 12 (2020), 209-265.  doi: 10.1142/S179352531950047X.  Google Scholar

[34]

A. F. Ritter, Topological quantum field theory structure on symplectic cohomology, J. Topol., 6 (2013), 391-489.  doi: 10.1112/jtopol/jts038.  Google Scholar

[35]

J. Robbin and D. Salamon, The Maslov index for paths, Topology, 32 (1993), 827-844.  doi: 10.1016/0040-9383(93)90052-W.  Google Scholar

[36]

S. Suhr and K. Zehmisch, Linking and closed orbits, Abh. Math. Semin. Univ. Hambg., 86 (2016), 133-150.  doi: 10.1007/s12188-016-0118-5.  Google Scholar

[37]

J. B. van den Berg, F. Pasquotto, T. Rot and R. C. A. M. Vandervorst, On periodic orbits in cotangent bundles of non-compact manifolds, J. Symplectic Geom., 14 (2016), 1145-1173.  doi: 10.4310/JSG.2016.v14.n4.a6.  Google Scholar

[38]

J. B. van den Berg, F. Pasquotto and R. C. Vandervorst, Closed characteristics on non-compact hypersurfaces in $\mathbb{R}^{2n}$, Math. Ann., 343 (2009), 247-284.  doi: 10.1007/s00208-008-0271-y.  Google Scholar

[39]

S. Venkatesh, Rabinowitz Floer homology and mirror symmetry, J. Topol., 11 (2018), 144-179.  doi: 10.1112/topo.12050.  Google Scholar

[40]

C. Viterbo, A proof of Weinstein's conjecture in $\mathbb{R}^{2n}$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987), 337-356.  doi: 10.1016/S0294-1449(16)30363-8.  Google Scholar

[41]

J. J. Wiśniewska, Rabinowitz Floer Homology for Tentacular Hamiltonians, Ph.D thesis, Vrije Universiteit Amsterdam, 2017. Google Scholar

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