Geodesic Completeness for Sobolev $H^{s}$-metrics on the Diffeomorphisms Group of the Circle

TL;DR

This paper proves that the diffeomorphism group of the circle equipped with a fractional Sobolev $H^s$ metric is geodesically complete when $s>3/2$, extending understanding of geometric properties of infinite-dimensional groups.

Contribution

It establishes geodesic completeness for the $H^s$ metric on the circle's diffeomorphism group for $s>3/2$, a significant result in infinite-dimensional geometry.

Findings

Geodesic completeness holds for $s>3/2$

Weak $H^s$ metric induces a complete Riemannian structure

Advances understanding of geometric analysis on diffeomorphism groups

Abstract

We prove that the weak Riemannian metric induced by the fractional Sobolev norm $H^s$ on the diffeomorphisms group of the circle is geodesically complete, provided $s>3/2$.