American Institute of Mathematical Sciences

Advanced
March  2016, 8(1): 1-12. doi: 10.3934/jgm.2016.8.1

Fredholm properties of the $L^{2}$ exponential map on the symplectomorphism group

James Benn 1,

1. 

3 Hardie Street, Palmerston North, Hokowhitu, 4410, New Zealand

Received  December 2014 Revised  December 2015 Published  February 2016

Let $M$ be a closed symplectic manifold with compatible symplectic form and Riemannian metric $g$. Here it is shown that the exponential mapping of the weak $L^{2}$ metric on the group of symplectic diffeomorphisms of $M$ is a non-linear Fredholm map of index zero. The result provides an interesting contrast between the $L^{2}$ metric and Hofer's metric as well as an intriguing difference between the $L^{2}$ geometry of the symplectic diffeomorphism group and the volume-preserving diffeomorphism group.
Keywords: conjugate point, geodesic, Fredholm map, symplectic Euler equations., Diffeomorphism group, Maxwell-Vlasov.
Mathematics Subject Classification: Primary: 35Q05, 35Q83, 47H99, 58C99, 37K65; Secondary: 82D1.
Citation: James Benn. Fredholm properties of the $L^{2}$ exponential map on the symplectomorphism group. Journal of Geometric Mechanics, 2016, 8 (1) : 1-12. doi: 10.3934/jgm.2016.8.1
References:
[1]

V. I. Arnold, On the Differential Geometry of Infinite-Dimensional Lie Groups and its appliction to the Hydrodynamics of Perfect Fluids,, Vladimir I. Arnold: Collected Works, (2014).   Google Scholar

[2]

V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics,, Springer-Verlag, (1998).   Google Scholar

[3]

D. Bao, J. Lafontaine and T. Ratiu, On a non-linear equation related to the geometry of the diffeomorphism group,, Pacific Journal of Mathematics, 158 (1993), 223.  doi: 10.2140/pjm.1993.158.223.  Google Scholar

[4]

D. Ebin, Geodesics on the symplectomorphism group,, Journal of Geometric and Functional Analysis, 22 (2012), 202.  doi: 10.1007/s00039-012-0150-2.  Google Scholar

[5]

D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid,, Annals of Mathematics, 92 (1970), 102.  doi: 10.2307/1970699.  Google Scholar

[6]

D. Ebin, G. Misiołek and S. Preston, Singularities of the exponential map on the volume-preserving diffeomorphism group,, Journal of Geometric and Functional Analysis, 16 (2006), 850.  doi: 10.1007/s00039-006-0573-8.  Google Scholar

[7]

D. Holm and C. Tronci, The geodesic vlasov equation and its integrable moment closures,, Journal of Geometric Mechanics, 1 (2009), 181.  doi: 10.3934/jgm.2009.1.181.  Google Scholar

[8]

H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics,, Springer, (1994).  doi: 10.1007/978-3-0348-8540-9.  Google Scholar

[9]

T. Kato, Perturbation Theory for Linear Operators,, Springer-Verlag, (1966).   Google Scholar

[10]

J. Marsden and A. Weinstein, The Hamiltonian structure of the Maxwell-Vlasov equations,, Physica D, 4 (1982), 394.  doi: 10.1016/0167-2789(82)90043-4.  Google Scholar

[11]

J. Marsden, A. Weinstein, R. Schmid and R. Spencer, Hamiltonian systems and symmetry groups with applications to plasma physics,, Atti della Accademia delle Scienze di Torino. Classe di Scienze Fisiche, 117 (1983), 289.   Google Scholar

[12]

G. Misiolek, Stability of flows of ideal fluids and the geometry of the groups of diffeomorphisms,, Indiana University Mathematics Journal, 42 (1993), 215.  doi: 10.1512/iumj.1993.42.42011.  Google Scholar

[13]

G. Misiołek, Conjugate points in $\mathcalD_{\mu}(\mathbbT^{2})$,, Proceedings of the American Mathematics Society, 124 (1996), 977.  doi: 10.1090/S0002-9939-96-03149-8.  Google Scholar

[14]

G. Misiołek and S. Preston, Fredholm properties of riemannian exponential maps on diffeomorphism groups,, Inventiones Mathematicae, 179 (2010), 191.  doi: 10.1007/s00222-009-0217-3.  Google Scholar

[15]

C. B. Morrey, Multiple Integrals in the Calculus of Variations,, Springer-Verlag, (1966).   Google Scholar

[16]

P. J. Morrison, The Maxwell-Vlasov equations as a continuous Hamiltonian system,, Physics Letters A, 80 (1980), 383.  doi: 10.1016/0375-9601(80)90776-8.  Google Scholar

[17]

S. Preston, For ideal fluids, Eulerian and Lagrangian instabilities are equivalent,, Journal of Geometric and Functional Analysis, 14 (2004), 1044.  doi: 10.1007/s00039-004-0482-7.  Google Scholar

[18]

S. Preston, On the volumorphism group, the first conjugate point is the hardest,, Communications in Mathematical Physics, 267 (2006), 493.  doi: 10.1007/s00220-006-0070-9.  Google Scholar

[19]

T. Ratiu and R. Schmid, Three remarkable diffeomorphism groups,, Mathematische Zeitschrift, 177 (1981), 81.  doi: 10.1007/BF01214340.  Google Scholar

[20]

R. Schmid, Infinite dimensional lie groups and algebras in mathematical physics,, Hindawi Advances in Mathematical Physics, 2010 (2010).   Google Scholar

[21]

A. Shnirelman, Generalized fluid flows, their approximations and applications,, Journal of Geometric and Functional Analysis, 4 (1994), 586.  doi: 10.1007/BF01896409.  Google Scholar

[22]

S. Smale, An infinite dimensional version of Sard's Theorem,, American Journal of Mathematics, 87 (1965), 861.  doi: 10.2307/2373250.  Google Scholar

[23]

N. K. Smolentsev, A Biinvariant Metric on the Group of Symplectic Diffeomorphisms and the Equation $\partial_t\DeltaF = {\DeltaF,F}$,, Translated from Sibirskii Matematicheskii Shurnal, 27 (1986), 150.   Google Scholar

[24]

M. Taylor, Partial Differential Equations I, Basic Theory,, Springer, (2011).  doi: 10.1007/978-1-4419-7055-8.  Google Scholar

[25]

I. Ustilovsky, Conjugate points on geodesics of hofer's metric,, Differential Geometry and its Applications, 6 (1996), 327.  doi: 10.1016/S0926-2245(96)00027-7.  Google Scholar

show all references

References:
[1]

V. I. Arnold, On the Differential Geometry of Infinite-Dimensional Lie Groups and its appliction to the Hydrodynamics of Perfect Fluids,, Vladimir I. Arnold: Collected Works, (2014).   Google Scholar

[2]

V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics,, Springer-Verlag, (1998).   Google Scholar

[3]

D. Bao, J. Lafontaine and T. Ratiu, On a non-linear equation related to the geometry of the diffeomorphism group,, Pacific Journal of Mathematics, 158 (1993), 223.  doi: 10.2140/pjm.1993.158.223.  Google Scholar

[4]

D. Ebin, Geodesics on the symplectomorphism group,, Journal of Geometric and Functional Analysis, 22 (2012), 202.  doi: 10.1007/s00039-012-0150-2.  Google Scholar

[5]

D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid,, Annals of Mathematics, 92 (1970), 102.  doi: 10.2307/1970699.  Google Scholar

[6]

D. Ebin, G. Misiołek and S. Preston, Singularities of the exponential map on the volume-preserving diffeomorphism group,, Journal of Geometric and Functional Analysis, 16 (2006), 850.  doi: 10.1007/s00039-006-0573-8.  Google Scholar

[7]

D. Holm and C. Tronci, The geodesic vlasov equation and its integrable moment closures,, Journal of Geometric Mechanics, 1 (2009), 181.  doi: 10.3934/jgm.2009.1.181.  Google Scholar

[8]

H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics,, Springer, (1994).  doi: 10.1007/978-3-0348-8540-9.  Google Scholar

[9]

T. Kato, Perturbation Theory for Linear Operators,, Springer-Verlag, (1966).   Google Scholar

[10]

J. Marsden and A. Weinstein, The Hamiltonian structure of the Maxwell-Vlasov equations,, Physica D, 4 (1982), 394.  doi: 10.1016/0167-2789(82)90043-4.  Google Scholar

[11]

J. Marsden, A. Weinstein, R. Schmid and R. Spencer, Hamiltonian systems and symmetry groups with applications to plasma physics,, Atti della Accademia delle Scienze di Torino. Classe di Scienze Fisiche, 117 (1983), 289.   Google Scholar

[12]

G. Misiolek, Stability of flows of ideal fluids and the geometry of the groups of diffeomorphisms,, Indiana University Mathematics Journal, 42 (1993), 215.  doi: 10.1512/iumj.1993.42.42011.  Google Scholar

[13]

G. Misiołek, Conjugate points in $\mathcalD_{\mu}(\mathbbT^{2})$,, Proceedings of the American Mathematics Society, 124 (1996), 977.  doi: 10.1090/S0002-9939-96-03149-8.  Google Scholar

[14]

G. Misiołek and S. Preston, Fredholm properties of riemannian exponential maps on diffeomorphism groups,, Inventiones Mathematicae, 179 (2010), 191.  doi: 10.1007/s00222-009-0217-3.  Google Scholar

[15]

C. B. Morrey, Multiple Integrals in the Calculus of Variations,, Springer-Verlag, (1966).   Google Scholar

[16]

P. J. Morrison, The Maxwell-Vlasov equations as a continuous Hamiltonian system,, Physics Letters A, 80 (1980), 383.  doi: 10.1016/0375-9601(80)90776-8.  Google Scholar

[17]

S. Preston, For ideal fluids, Eulerian and Lagrangian instabilities are equivalent,, Journal of Geometric and Functional Analysis, 14 (2004), 1044.  doi: 10.1007/s00039-004-0482-7.  Google Scholar

[18]

S. Preston, On the volumorphism group, the first conjugate point is the hardest,, Communications in Mathematical Physics, 267 (2006), 493.  doi: 10.1007/s00220-006-0070-9.  Google Scholar

[19]

T. Ratiu and R. Schmid, Three remarkable diffeomorphism groups,, Mathematische Zeitschrift, 177 (1981), 81.  doi: 10.1007/BF01214340.  Google Scholar

[20]

R. Schmid, Infinite dimensional lie groups and algebras in mathematical physics,, Hindawi Advances in Mathematical Physics, 2010 (2010).   Google Scholar

[21]

A. Shnirelman, Generalized fluid flows, their approximations and applications,, Journal of Geometric and Functional Analysis, 4 (1994), 586.  doi: 10.1007/BF01896409.  Google Scholar

[22]

S. Smale, An infinite dimensional version of Sard's Theorem,, American Journal of Mathematics, 87 (1965), 861.  doi: 10.2307/2373250.  Google Scholar

[23]

N. K. Smolentsev, A Biinvariant Metric on the Group of Symplectic Diffeomorphisms and the Equation $\partial_t\DeltaF = {\DeltaF,F}$,, Translated from Sibirskii Matematicheskii Shurnal, 27 (1986), 150.   Google Scholar

[24]

M. Taylor, Partial Differential Equations I, Basic Theory,, Springer, (2011).  doi: 10.1007/978-1-4419-7055-8.  Google Scholar

[25]

I. Ustilovsky, Conjugate points on geodesics of hofer's metric,, Differential Geometry and its Applications, 6 (1996), 327.  doi: 10.1016/S0926-2245(96)00027-7.  Google Scholar

[1]

Darryl D. Holm, Cesare Tronci. Geodesic Vlasov equations and their integrable moment closures. Journal of Geometric Mechanics, 2009, 1 (2) : 181-208. doi: 10.3934/jgm.2009.1.181
[2]

Jianwei Yang, Ruxu Lian, Shu Wang. Incompressible type euler as scaling limit of compressible Euler-Maxwell equations. Communications on Pure & Applied Analysis, 2013, 12 (1) : 503-518. doi: 10.3934/cpaa.2013.12.503
[3]

Christian Bonatti, Stanislav Minkov, Alexey Okunev, Ivan Shilin. Anosov diffeomorphism with a horseshoe that attracts almost any point. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 441-465. doi: 10.3934/dcds.2020017
[4]

Björn Sandstede, Arnd Scheel. Relative Morse indices, Fredholm indices, and group velocities. Discrete & Continuous Dynamical Systems - A, 2008, 20 (1) : 139-158. doi: 10.3934/dcds.2008.20.139
[5]

Joachim Escher, Rossen Ivanov, Boris Kolev. Euler equations on a semi-direct product of the diffeomorphisms group by itself. Journal of Geometric Mechanics, 2011, 3 (3) : 313-322. doi: 10.3934/jgm.2011.3.313
[6]

Joachim Escher, Boris Kolev. Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle. Journal of Geometric Mechanics, 2014, 6 (3) : 335-372. doi: 10.3934/jgm.2014.6.335
[7]

Juan Campos, Rafael Obaya, Massimo Tarallo. Recurrent equations with sign and Fredholm alternative. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 959-977. doi: 10.3934/dcdss.2016036
[8]

Renjun Duan, Shuangqian Liu. Cauchy problem on the Vlasov-Fokker-Planck equation coupled with the compressible Euler equations through the friction force. Kinetic & Related Models, 2013, 6 (4) : 687-700. doi: 10.3934/krm.2013.6.687
[9]

Sergiu Klainerman, Gigliola Staffilani. A new approach to study the Vlasov-Maxwell system. Communications on Pure & Applied Analysis, 2002, 1 (1) : 103-125. doi: 10.3934/cpaa.2002.1.103
[10]

George Papadopoulos, Holger R. Dullin. Semi-global symplectic invariants of the Euler top. Journal of Geometric Mechanics, 2013, 5 (2) : 215-232. doi: 10.3934/jgm.2013.5.215
[11]

Yue-Jun Peng, Shu Wang. Asymptotic expansions in two-fluid compressible Euler-Maxwell equations with small parameters. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 415-433. doi: 10.3934/dcds.2009.23.415
[12]

Min Li, Xueke Pu, Shu Wang. Quasineutral limit for the compressible two-fluid Euler–Maxwell equations for well-prepared initial data. Electronic Research Archive, 2020, 28 (2) : 879-895. doi: 10.3934/era.2020046
[13]

Pablo D. Carrasco, Túlio Vales. A symmetric Random Walk defined by the time-one map of a geodesic flow. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020390
[14]

Feride Tığlay. Integrating evolution equations using Fredholm determinants. Electronic Research Archive, 2021, 29 (2) : 2141-2147. doi: 10.3934/era.2020109
[15]

Marie-Claude Arnaud. A nondifferentiable essential irrational invariant curve for a $C^1$ symplectic twist map. Journal of Modern Dynamics, 2011, 5 (3) : 583-591. doi: 10.3934/jmd.2011.5.583
[16]

Benjamin Boutin, Frédéric Coquel, Philippe G. LeFloch. Coupling techniques for nonlinear hyperbolic equations. Ⅱ. resonant interfaces with internal structure. Networks & Heterogeneous Media, 2021  doi: 10.3934/nhm.2021007
[17]

David Ralston, Serge Troubetzkoy. Ergodic infinite group extensions of geodesic flows on translation surfaces. Journal of Modern Dynamics, 2012, 6 (4) : 477-497. doi: 10.3934/jmd.2012.6.477
[18]

Raz Kupferman, Asaf Shachar. On strain measures and the geodesic distance to $SO_n$ in the general linear group. Journal of Geometric Mechanics, 2016, 8 (4) : 437-460. doi: 10.3934/jgm.2016015
[19]

John Franks, Michael Handel. Some virtually abelian subgroups of the group of analytic symplectic diffeomorphisms of a surface. Journal of Modern Dynamics, 2013, 7 (3) : 369-394. doi: 10.3934/jmd.2013.7.369
[20]

Martin Pinsonnault. Maximal compact tori in the Hamiltonian group of 4-dimensional symplectic manifolds. Journal of Modern Dynamics, 2008, 2 (3) : 431-455. doi: 10.3934/jmd.2008.2.431

2019 Impact Factor: 0.649

Tools

Metrics

  • PDF downloads (101)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences