A family of non-simple surfaces whose transport twistor spaces admit global blow-down maps
François Monard, Zhengyi Qi
TL;DR
The paper constructs an explicit non-simple, convex, non-trapping family of surfaces whose Transport Twistor spaces admit global holomorphic blow-down maps, extending beyond previously known simple near-constant-curvature cases. For the model disk $M={\mathbb D}_R$ with metric $g_\kappa$, it defines an explicit holomorphic map $\beta=(w,\xi)$ in coordinates $(z,\nu)$, with $w=(z-\bar z\nu^2) e^{\frac{\kappa(z\bar z-\bar z^2\nu^2)}{2}}$ and $\xi=\nu e^{\frac{\kappa(z\bar z-\bar z^2\nu^2)}{2}}$, and proves that this map provides holomorphic blow-down structure, hence $Z^\circ$ embeds biholomorphically into $\mathbb{C}^2$ for all admissible $(R,\kappa)$. The work also analyzes consequences for geodesically invariant distributions and the attenuated geodesic X-ray problem, linking invariant functions on $SM$ to holomorphic data on $Z$ and presenting a non-simple analogue of a key exact-sequence result. A sharp simplicity criterion is obtained via the scattering function, showing simplicity iff $|\kappa|R^2<1$, while conjugate points occur for $\kappa R^2\ge 1$. These results open the door to constructing and perturbing blow-down structures in non-simple geometries and have potential implications for tensor tomography and inverse problems on surfaces with boundary.
Abstract
In the literature on X-ray transform and Transport Twistor (TT) spaces, blow-down maps (or maps with holomorphic blow-down structure as defined in [BMP24]) are maps that desingularize the degenerate complex structure of the TT space of an oriented Riemannian surface, while collapsing (yet separating) geodesics of the unit tangent bundle of that surface. Such maps were originally constructed in [BMP24] for near-constant curvature simple surfaces, showing that the interior of their TT space is biholomorphic to an open set in standard $\mathbb{C}^2$. The construction there relied on a microlocal argument leveraging the absence of conjugate points. In this note, we construct an explicit example of a family of convex, non-trapping Riemannian surfaces, some of which have conjugate points, yet all of whose TT spaces admit a global blow-down map. We also discuss a consequence on the existence of special geodesically invariant functions and its application to geometric inverse problems.