Local and global blow-downs of transport twistor space

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 Z→ C² 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.