Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 045, 45 pages      arXiv:2403.05985      https://doi.org/10.3842/SIGMA.2026.045

Local and Global Blow Downs of Transport Twistor Space

Jan Bohr a, François Monard b and Gabriel P. Paternain c
a) Mathematical Institute, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany
b) Department of Mathematics, University of California, Santa Cruz, CA 95064, USA
c) Department of Mathematics, University of Washington, Seattle, WA 98195, USA

Received April 02, 2025, in final form April 20, 2026; Published online May 05, 2026

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 $\beta\colon Z\to \mathbb{C}^2$ 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 $\beta$-maps for constant curvature metrics and their perturbations and local $\beta$-maps for arbitrary metrics, thereby proving a version of the classical Newlander-Nirenberg theorem for degenerate complex structures.

Key words: transport twistor space; geometric inverse problems; holomorphic blow down structure; geodesic X-ray transform.

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