Spheres, Kähler geometry and the Hunter-Saxton system (Q2831277)

This article deals with a generalization of the Hunter-Saxton (HS) system. Recall that the basic HS equation can be seen both as the equation governing waves in shallow water and as the geodesic flow on a Lie group of diffeomorphisms (on \(S^1\), with a prescribed fixed point and a specific metric, for the details see [\textit{B. Khesin} and \textit{G. Misiołek}, Adv. Math. 176, 1, 116--144 (2003; Zbl 1017.37039)]).NEWLINENEWLINEThe system studied here is a two-component integrable generalization of HS (we will call it 2HS) which describes both waves in the shallow water regime with nonzero constant vorticity [\textit{J. Escher} et al., Discrete Contin. Dyn. Syst. 19, No. 3, 493--513 (2007; Zbl 1149.35307)] and the geodesic flow on a semidirect product Lie group (of sufficiently regular diffeomorphisms that fix a point).NEWLINENEWLINEThe main result shows, roughly, that the semidirect product above can be turned into a weak Riemannian manifold which is isometric to a subset of the unit sphere in \(L^2(S^1,\mathbb{C})\): the proof exhibits an explicit map which is a diffeomorphism and an isometry; thus the curvature is computed explicitly by Gauss-Codazzi. This isometry allows the authors to characterize both type of solutions of the 2HS system, i.e., those that have a blowup time and those that exist globally.NEWLINENEWLINEMoreover, for the 2HS system, if we require that the second component has mean value zero, then the semidirect product constructed is a Kähler manifold: the needed properties are established either by direct methods or by using the arguments of [\textit{D. G. Ebin} and \textit{J. Marsden}, Ann. Math. (2) 92, 102--163 (1970; Zbl 0211.57401)]. Last, it is shown that the metric on this manifold is isometric to the Fubini-Study one, thus it can be seen as a subset of the complex projective space, also providing an example of an infinite-dimensional Hopf fibration.