Can we run to infinity? The diameter of the diffeomorphism group with respect to right-invariant Sobolev metrics

Abstract

The group \({\text {Diff}}({\mathcal {M}})\) of diffeomorphisms of a closed manifold \({\mathcal {M}}\) is naturally equipped with various right-invariant Sobolev norms \(W^{s,p}\). Recent work showed that for sufficiently weak norms, the geodesic distance collapses completely (namely, when \(sp\le \dim {\mathcal {M}}\) and \(s<1\)). But when there is no collapse, what kind of metric space is obtained? In particular, does it have a finite or infinite diameter? This is the question we study in this paper. We show that the diameter is infinite for strong enough norms, when \((s-1)p\ge \dim {\mathcal {M}}\), and that for spheres the diameter is finite when \((s-1)p<1\). In particular, this gives a full characterization of the diameter of \({\text {Diff}}(S^1)\). In addition, we show that for \({\text {Diff}}_c({\mathbb {R}}^n)\), if the diameter is not zero, it is infinite.