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 PSL₂(ℝ) \ 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.

Keywords:

• Primary 58B20, 58D15, 58E40
• Teichons
• diffeomorphism group
• Weil–Peterson metric
• universal Teichmüller space
• EPDiff