Geodesic Flows on Diffeomorphisms of the Circle, Grassmannians, and the Geometry of the Periodic KdV Equation

TL;DR

This paper constructs a geometric framework for the space of circle diffeomorphisms, embedding it into an infinite-dimensional Grassmannian, and uses this to analyze the periodic KdV equation's solutions and their long-term behavior.

Contribution

It introduces a Hilbert manifold of circle diffeomorphisms with a Kaehler metric, embedding it into a Grassmannian, and links its geodesics to solutions of the periodic KdV equation.

Findings

Sectional curvature of the manifold is negative in holomorphic directions.

Geodesics exist for all time, ensuring long-term solutions.

Long-time existence of KdV solutions with specific initial data.

Abstract

We start by constructing a Hilbert manifold T of orientation preserving diffeomorphisms of the circle (modulo the group of bi-holomorphic self-mappings of the disc). This space, which could be thought of as a completion of the universal Teichmueller space, is endowed with a right-invariant Kaehler metric. Using results from the theory of quasiconformal mappings we construct an embedding of T into the infinite dimensional Segal-Wilson Grassmannian. The latter turns out to be a very natural ambient space for T. This allows us to prove that T's sectional curvature is negative in the holomorphic directions and by a reasoning along the lines of Cartan-Hadamard's theory that its geodesics exist for all time. The geodesics of T lead to solutions of the periodic Korteweg-de Vries (KdV) equation by means of V. Arnold's generalization of Euler's equation. As an application, we obtain long-time existence of solutions to the periodic KdV equation with initial data in a certain closed subspace of the periodic Sobolev space of index 3/2.