Approximating the Weil--Petersson Metric Geodesics on the Universal Teichmüller Space by Singular Solutions

Abstract

We propose and investigate a numerical shooting method for computing geodesics in the Weil--Petersson (WP) metric on the universal Teichmuller space $T(1)$. This space, or rather the coset subspace ${{PSL}}_2(\mathbb{R})\backslash{{Diff}}(S^1)$, has another realization as the space of smooth, simple closed planar curves modulo translations and scalings. This alternate identification of $T(1)$ is a convenient metrization of the space of shapes and provides an immediate application for our algorithm in computer vision. The geodesic equation on $T(1)$ with the WP metric is EPDiff($S^1$), the Euler--Poincare equation on the group of diffeomorphisms of the circle $S^1$, and admits a class of soliton-like solutions named teichons [S. Kushnarev, Experiment. Math., 18 (2009), pp. 325--336]. Our method relies on approximating the geodesic with these teichon solutions, which have momenta given by a finite linear combination of delta functions. The geodesic equation for this simpler set of solutions is more tractabl...