# Geodesic completeness of the $H^{3/2}$ metric on $\mathrm{Diff}(S^{1})$

## TL;DR

This paper investigates the long-time existence of solutions to the Euler--Arnold equation for the $H^{3/2}$-metric on the circle diffeomorphism group, filling a gap in understanding at the critical Sobolev index.

## Contribution

It proves long-time existence results for the $H^{3/2}$-metric at the critical index, extending previous work that covered higher Sobolev orders.

## Key findings

- Long-time solutions exist at the critical Sobolev index $s=3/2$.
- The $H^{3/2}$-metric does not induce a complete Riemannian metric.
- Behavior differs from higher-order Sobolev metrics.

## Abstract

Of concern is the study of the long-time existence of solutions to the Euler--Arnold equation of the right-invariant $H^{3/2}$-metric on the diffeomorphism group of the circle. In previous work by Escher and Kolev it has been shown that this equation admits long-time solutions if the order $s$ of the metric is greater than $3/2$, the behaviour for the critical Sobolev index $s=3/2$ has been left open. In this article we fill this gap by proving the analogous result also for the boundary case. The behaviour of the $H^{3/2}$-metric is, however, still different from its higher order counter parts, as it does not induce a complete Riemannian metric on any group of Sobolev diffeomorphisms.