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

James  Benn

doi:10.3934/jgm.2016.8.1

Article Contents

2016, Volume 8, Issue 1: 1-12. Doi: 10.3934/jgm.2016.8.1

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

James Benn^1,

1.
3 Hardie Street, Palmerston North, Hokowhitu, 4410

Received: November 30, 2014

Revised: November 30, 2015

Published: February 2016

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Abstract

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.

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Mathematics Subject Classification: Primary: 35Q05, 35Q83, 47H99, 58C99, 37K65; Secondary: 82D10.

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