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2009 Teichons: Solitonlike Geodesics on Universal Teichmüller Space
Sergey Kushnarev
Experiment. Math. 18(3): 325-336 (2009).


This paper studies $\EPDiff(S^1)$, the Euler-Poincaré equation for diffeomorphisms of $S^1$, with the Weil-Petersson metric on the coset space $\PSL_2(\R)\setminus\Diff(S^1)$. This coset space is known as the universal Teichmüller space. It has another realization as the space of smooth simple closed curves modulo translations and scalings. $\EPDiff(S^1)$ admits a class of solitonlike solutions (teichons) in which the ``momentum'' $m$ is a distribution. The solutions of this equation can also be thought of as paths in the space of simple closed plane curves that minimize a certain energy. In this paper we study the solution in the special case that $m$ is expressed as a sum of four delta functions. We prove the existence of the solution for infinite time and find bounds on its long-term behavior, showing that it is asymptotic to a one-parameter subgroup in $\Diff(S^1)$. We then present a series of numerical experiments on solitons with more delta functions and make some conjectures about these.


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Sergey Kushnarev. "Teichons: Solitonlike Geodesics on Universal Teichmüller Space." Experiment. Math. 18 (3) 325 - 336, 2009.


Published: 2009
First available in Project Euclid: 25 November 2009

zbMATH: 1185.58003
MathSciNet: MR2555702

Primary: 58B20 , 58D15 , 58E40

Keywords: diffeomorphism group , EPDiff , Teichons , universal Teichmüller space , Weil--Peterson metric

Rights: Copyright © 2009 A K Peters, Ltd.

Vol.18 • No. 3 • 2009
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