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

Abstract

Of concern is the study of the long-time existence of solutions to the Euler–Arnold equation of the right-invariant \(H^{\frac{3}{2}}\)-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 \(\frac{3}{2}\), but the behaviour for the critical Sobolev index \(s=\frac{3}{2}\) has been left open. In this article we fill this gap by proving the analogous result also for the boundary case. We show that the behaviour is the same for all Sobolev metrics of order \(\frac{3}{2}\) regardless of lower-order terms.