Local and global blow-downs of transport twistor space

TL;DR

This work develops holomorphic blow-down maps β from transport twistor spaces Z of oriented Riemannian surfaces to C^2 to desingularise the naturally degenerate complex structure. It proves global β-maps for constant-curvature disks and their small perturbations, and establishes local β-maps for arbitrary metrics, yielding a transport Newlander–Nirenberg theorem in a degenerate setting. The construction hinges on canonical β-extensions tied to the geodesic X-ray transform and its normal operator, with perturbation theory backed by Grubb–Hörmander-type analysis and boundary-structure control via the C_α^∞-framework. The results lead to identity principles for holomorphic curves and maps into Z, supporting rigidity statements for biholomorphisms and establishing a robust link between complex-analytic methods and geometric inverse problems in two dimensions.

Abstract

Transport twistor spaces are degenerate complex $2$-dimensional manifolds $Z$ that complexify transport problems on Riemannian surfaces, appearing e.g.~in geometric inverse problems. This article considers maps $β\colon Z\to \mathbb{C}^2$ with a holomorphic blow-down structure that resolve the degeneracy of the complex structure and allow to gain insight into the complex geometry of $Z$. The main theorems provide global $β$-maps for constant curvature metrics and their perturbations and local $β$-maps for arbitrary metrics, thereby proving a version of the classical Newlander--Nirenberg theorem for degenerate complex structures.

Theorems & Definitions (96)

• Theorem
• Remark 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6: Identity principle for holomorphic curves
• Corollary 1.7: Identity principle for holomorphic maps
• Definition 2.1
• Definition 2.2