The Transport Oka-Grauert Principle for Simple Surfaces

TL;DR

This paper establishes a transport version of the Oka-Grauert principle for simple surfaces, linking matrix attenuations to holomorphic vector bundles and solving an open problem in the field.

Contribution

It introduces a novel twistor correspondence for Riemannian surfaces, proving the absence of nontrivial holomorphic vector bundles on the twistor space of simple surfaces.

Findings

Supports no nontrivial holomorphic vector bundles on the twistor space of simple surfaces

Provides a range characterization for the non-Abelian X-ray transform

Solves an open problem on matrix holomorphic integrating factors

Abstract

This article considers the attenuated transport equation on Riemannian surfaces in the light of a novel twistor correspondence under which matrix attenuations correspond to holomorphic vector bundles on a complex surface. The main result is a transport version of the classical Oka-Grauert principle and states that the twistor space of a simple surface supports no nontrivial holomorphic vector bundles. This solves an open problem on the existence of matrix holomorphic integrating factors on simple surfaces and is applied to give a range characterisation for the non-Abelian X-ray transform. The main theorem is proved using the inverse function theorem of Nash and Moser and the required tame estimates are obtained from recent results on the injectivity of attenuated X-ray transforms and microlocal analysis of the associated normal operators.