Tomography of 1-forms on a gas giant

TL;DR

The paper proves solenoidal injectivity of the geodesic X-ray transform for smooth up to boundary 1-forms on non-trapping gas giant manifolds with non-positive curvature, showing that if $I f=0$ then $f$ is exact modulo a boundary gauge. The authors develop a cohesive framework around the gas giant geometry, with the metric $g=\rho^{-1}\bar{g}$ and a carefully chosen smooth structure on the cotangent bundle, enabling a well-behaved geodesic flow up to the boundary. A core contribution is combining boundary-determination via short geodesics, fine-regularity results for the integral function $u^f$, and a Pestov energy identity to conclude that $f$ must be $\mathrm{d}p$ for some boundary-vanishing potential $p$, up to a gauge near $\partial M$. This work advances tensor tomography in singular geometric settings between standard Riemannian and asymptotically hyperbolic geometries and provides a rigorous path to reconstructing flow information from Doppler-type travel-time data in gas giant models.

Abstract

We show that on gas giant manifolds the geodesic X-ray transform is solenoidally injective on one-forms that are smooth up to the boundary in an appropriate smooth structure. A gas giant manifold is a conformally blown up Riemannian manifold whose boundary singularity is milder than asymptotically hyperbolic. The proof is based on a Pestov identity and asymptotic analysis of short geodesics.

Theorems & Definitions (48)

• Theorem 1.1
• Definition 2.1: Function space $\Omega$
• Lemma 2.2: proved in section \ref{['sec:Boundary determination']}
• Lemma 2.3: proved in section \ref{['sec:Regularity of the integral function']}
• Lemma 2.4: proved in section \ref{['sec:Pestov identity']}
• proof : Proof of theorem \ref{['thm:injectivity-mod-gauge']}
• Proposition 4.1
• proof
• Proposition 4.2
• proof