Teichons: Solitonlike Geodesics on Universal Teichmüller Space
Kushnarev, Sergey
Experiment. Math., Tome 18 (2009) no. 1, p. 325-336 / Harvested from Project Euclid
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.
Publié le : 2009-05-15
Classification:  Teichons,  diffeomorphism group,  Weil--Peterson metric,  universal Teichmüller space,  EPDiff,  58B20,  58D15,  58E40
@article{1259158469,
     author = {Kushnarev, Sergey},
     title = {Teichons: Solitonlike Geodesics on Universal Teichm\"uller Space},
     journal = {Experiment. Math.},
     volume = {18},
     number = {1},
     year = {2009},
     pages = { 325-336},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1259158469}
}
Kushnarev, Sergey. Teichons: Solitonlike Geodesics on Universal Teichmüller Space. Experiment. Math., Tome 18 (2009) no. 1, pp.  325-336. http://gdmltest.u-ga.fr/item/1259158469/