# American Institute of Mathematical Sciences

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

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

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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.
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). [2] V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics,, Springer-Verlag, (1998). [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. [4] D. Ebin, Geodesics on the symplectomorphism group,, Journal of Geometric and Functional Analysis, 22 (2012), 202. doi: 10.1007/s00039-012-0150-2. [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. [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. [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. [8] H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics,, Springer, (1994). doi: 10.1007/978-3-0348-8540-9. [9] T. Kato, Perturbation Theory for Linear Operators,, Springer-Verlag, (1966). [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. [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. [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. [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. [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. [15] C. B. Morrey, Multiple Integrals in the Calculus of Variations,, Springer-Verlag, (1966). [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. [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. [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. [19] T. Ratiu and R. Schmid, Three remarkable diffeomorphism groups,, Mathematische Zeitschrift, 177 (1981), 81. doi: 10.1007/BF01214340. [20] R. Schmid, Infinite dimensional lie groups and algebras in mathematical physics,, Hindawi Advances in Mathematical Physics, 2010 (2010). [21] A. Shnirelman, Generalized fluid flows, their approximations and applications,, Journal of Geometric and Functional Analysis, 4 (1994), 586. doi: 10.1007/BF01896409. [22] S. Smale, An infinite dimensional version of Sard's Theorem,, American Journal of Mathematics, 87 (1965), 861. doi: 10.2307/2373250. [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. [24] M. Taylor, Partial Differential Equations I, Basic Theory,, Springer, (2011). doi: 10.1007/978-1-4419-7055-8. [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.

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##### 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). [2] V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics,, Springer-Verlag, (1998). [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. [4] D. Ebin, Geodesics on the symplectomorphism group,, Journal of Geometric and Functional Analysis, 22 (2012), 202. doi: 10.1007/s00039-012-0150-2. [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. [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. [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. [8] H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics,, Springer, (1994). doi: 10.1007/978-3-0348-8540-9. [9] T. Kato, Perturbation Theory for Linear Operators,, Springer-Verlag, (1966). [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. [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. [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. [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. [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. [15] C. B. Morrey, Multiple Integrals in the Calculus of Variations,, Springer-Verlag, (1966). [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. [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. [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. [19] T. Ratiu and R. Schmid, Three remarkable diffeomorphism groups,, Mathematische Zeitschrift, 177 (1981), 81. doi: 10.1007/BF01214340. [20] R. Schmid, Infinite dimensional lie groups and algebras in mathematical physics,, Hindawi Advances in Mathematical Physics, 2010 (2010). [21] A. Shnirelman, Generalized fluid flows, their approximations and applications,, Journal of Geometric and Functional Analysis, 4 (1994), 586. doi: 10.1007/BF01896409. [22] S. Smale, An infinite dimensional version of Sard's Theorem,, American Journal of Mathematics, 87 (1965), 861. doi: 10.2307/2373250. [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. [24] M. Taylor, Partial Differential Equations I, Basic Theory,, Springer, (2011). doi: 10.1007/978-1-4419-7055-8. [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.
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