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arXiv · 1904.12523

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

Martin Bauer·Boris Kolev·Stephen Preston
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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.

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Martin Bauer, Boris Kolev, Stephen Preston. 2019-04-29. Geodesic completeness of the $H^{3/2}$ metric on $\mathrm{Diff}(S^{1})$. https://arxiv.org/abs/1904.12523

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