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.
Cet article étudie l’équation de transport atténuée sur les surfaces riemanniennes à la lumière d’une nouvelle correspondance de twisteurs dans laquelle les atténuations de matrice correspondent à des fibrés vectoriels holomorphes sur une surface complexe. Le résultat principal est une version de transport du principe classique d’Oka-Grauert et stipule que l’espace des twisteurs d’une surface simple ne supporte aucun fibré vectoriel holomorphe non trivial. Ceci résout un problème ouvert sur l’existence de facteurs intégrants holomorphes matriciels sur des surfaces simples et est appliqué pour donner une caractérisation du domaine pour la transformation en rayons X non abélienne. Le théorème principal est démontré en utilisant le théorème d’inversion locale de Nash et Moser, et les estimations nécessaires sont obtenues à partir de résultats récents sur l’injectivité des transformées en rayons X atténuées et l’analyse microlocale des opérateurs normaux associés.
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Keywords: Non-Abelian X-ray transform, holomorphic integrating factors, transport equation, Oka-Grauert principle
Mot clés : Transformation en rayons X non abélienne, facteurs intégrants holomorphes, équation de transport, principe d’Oka-Grauert
Jan Bohr 1; Gabriel P. Paternain 2
@article{JEP_2023__10__727_0, author = {Jan Bohr and Gabriel P. Paternain}, title = {The {Transport} {Oka-Grauert} principle for simple surfaces}, journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques}, pages = {727--769}, publisher = {\'Ecole polytechnique}, volume = {10}, year = {2023}, doi = {10.5802/jep.231}, language = {en}, url = {https://jep.centre-mersenne.org/articles/10.5802/jep.231/} }
TY - JOUR AU - Jan Bohr AU - Gabriel P. Paternain TI - The Transport Oka-Grauert principle for simple surfaces JO - Journal de l’École polytechnique — Mathématiques PY - 2023 SP - 727 EP - 769 VL - 10 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.231/ DO - 10.5802/jep.231 LA - en ID - JEP_2023__10__727_0 ER -
%0 Journal Article %A Jan Bohr %A Gabriel P. Paternain %T The Transport Oka-Grauert principle for simple surfaces %J Journal de l’École polytechnique — Mathématiques %D 2023 %P 727-769 %V 10 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.231/ %R 10.5802/jep.231 %G en %F JEP_2023__10__727_0
Jan Bohr; Gabriel P. Paternain. The Transport Oka-Grauert principle for simple surfaces. Journal de l’École polytechnique — Mathématiques, Volume 10 (2023), pp. 727-769. doi : 10.5802/jep.231. https://jep.centre-mersenne.org/articles/10.5802/jep.231/
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