Title: Geometry, spectral analysis and inverse problems on gas giants
Abstract: On gas giant planets the speed of sound is isotropic and goes to zero at the surface. Geometrically, this corresponds to a Riemannian manifold whose metric tensor has a conformal blow-up near the boundary. The blow-up is tamer than in asymptotically hyperbolic geometry: the boundary is at a finite distance. We study the basic geometry of gas giant Riemannian metrics, including properties of geodesics near the boundary, the Hausdorff dimension of the boundary, and discreteness of the spectrum of the Laplace–Beltrami operator. We present the spectral analysis of this operator and derive the Weyl law. The involved exponents depend on the Hausdorff dimension which, in the supercritical case (which is the relevant case for Jupiter and Saturn), is larger than the topological dimension. We solve two inverse problems for simple gas giant planets, proving that the metric is uniquely determined by its boundary distance data and that the geodesic ray transform is injective. The study of Weyl asymptotics, which reflects some properties of the singular metric and determines the blow-up in the supercitical case, is a preliminary step towards analyzing other inverse problems.
Joint research with Y. Colin de Verdìère, C. Dietze, J. Ilmavirta, A. Kykkänen, R. Mazzeo and E. Trélat.
