Teichons: Solitonlike Geodesics on Universal Teichmüller Space

Abstract

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.