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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, Vol. 2, Springer, New York, 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), 223-242. 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), 202-212. 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), 102-163. 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), 850-868. 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), 181-208. 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), 394-406. 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, Matematiche e Naturali, 117 (1983), 289-340.  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), 215-235. 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), 977-982. 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), 191-227. 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), 383-386. 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), 1044-1062. 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), 493-513. doi: 10.1007/s00220-006-0070-9.  Google Scholar

[19]

T. Ratiu and R. Schmid, Three remarkable diffeomorphism groups, Mathematische Zeitschrift, 177 (1981), 81-100. 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), Art. ID 280362, 35 pp.  Google Scholar

[21]

A. Shnirelman, Generalized fluid flows, their approximations and applications, Journal of Geometric and Functional Analysis, 4 (1994), 586-620. doi: 10.1007/BF01896409.  Google Scholar

[22]

S. Smale, An infinite dimensional version of Sard's Theorem, American Journal of Mathematics, 87 (1965), 861-866. 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), 150-156.  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), 327-342. 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, Vol. 2, Springer, New York, 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), 223-242. 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), 202-212. 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), 102-163. 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), 850-868. 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), 181-208. 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), 394-406. 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, Matematiche e Naturali, 117 (1983), 289-340.  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), 215-235. 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), 977-982. 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), 191-227. 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), 383-386. 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), 1044-1062. 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), 493-513. doi: 10.1007/s00220-006-0070-9.  Google Scholar

[19]

T. Ratiu and R. Schmid, Three remarkable diffeomorphism groups, Mathematische Zeitschrift, 177 (1981), 81-100. 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), Art. ID 280362, 35 pp.  Google Scholar

[21]

A. Shnirelman, Generalized fluid flows, their approximations and applications, Journal of Geometric and Functional Analysis, 4 (1994), 586-620. doi: 10.1007/BF01896409.  Google Scholar

[22]

S. Smale, An infinite dimensional version of Sard's Theorem, American Journal of Mathematics, 87 (1965), 861-866. 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), 150-156.  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), 327-342. doi: 10.1016/S0926-2245(96)00027-7.  Google Scholar

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