The Transport Oka-Grauert principle for simple surfaces

Jan Bohr; Gabriel P. Paternain

doi:10.5802/jep.231

The Transport Oka-Grauert principle for simple surfaces
[Le principe de transport d’Oka-Grauert pour les surfaces simples]

Jan Bohr ¹ ; Gabriel P. Paternain ²

¹ Mathematical Institute of the University of Bonn Endenicher Allee 60, 53115 Bonn, Germany
² Department of Pure Mathematics and Mathematical Statistics, University of Cambridge Cambridge CB3 0WB, UK

Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 727-769.

Résumé
Abstract

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.

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.

Reçu le : 2022-04-18
Accepté le : 2023-04-11
Publié le : 2023-05-04

DOI : 10.5802/jep.231

Classification : 53C65, 58J40, 32Q28
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

Affiliations des auteurs :

Jan Bohr ¹ ; Gabriel P. Paternain ²

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@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, Tome 10 (2023), pp. 727-769. doi : 10.5802/jep.231. https://jep.centre-mersenne.org/articles/10.5802/jep.231/

Bibliographie
Cité par

[1] G. Ainsworth & Y. M. Assylbekov - “On the range of the attenuated magnetic ray transform for connections and Higgs fields”, Inverse Probl. Imaging 9 (2015) no. 2, p. 317-335 | DOI | MR | Zbl

[2] Y. M. Assylbekov, F. Monard & G. Uhlmann - “Inversion formulas and range characterizations for the attenuated geodesic ray transform”, J. Math. Pures Appl. (9) 111 (2018), p. 161-190 | DOI | MR | Zbl

[3] M. F. Atiyah, N. J. Hitchin & I. M. Singer - “Self-duality in four-dimensional Riemannian geometry”, Proc. Roy. Soc. London Ser. A 362 (1978) no. 1711, p. 425-461 | DOI | MR | Zbl

[4] J. Bohr - Stability, range and statistical aspects of non-Abelian X-ray tomography, Ph. D. Thesis, University of Cambridge, 2022, available at https://www.repository.cam.ac.uk/handle/1810/342060

[5] J. Bohr, T. Lefeuvre & G. P. Paternain - “Invariant distributions and the transport twistor space of closed surfaces”, 2023 | arXiv

[6] J. Bohr & R. Nickl - “On log-concave approximations of high-dimensional posterior measures and stability properties in non-linear inverse problems”, 2021, to appear in Ann. Inst. H. Poincaré Probab. Statist. | arXiv

[7] N. M. Desai, W. R. B. Lionheart, M. Sales, M. Strobl & S. Schmidt - “Polarimetric neutron tomography of magnetic fields: uniqueness of solution and reconstruction”, Inverse Problems 36 (2020) no. 4, p. 045001, 17 | DOI | MR | Zbl

[8] S. K. Donaldson & P. B. Kronheimer - The geometry of four-manifolds, Oxford Math. Monographs, The Clarendon Press, Oxford University Press, New York, 1990

[9] M. Dubois-Violette - “Structures complexes au-dessus des variétés, applications”, in Mathematics and physics (Paris, 1979/1982), Progress in Math., vol. 37, Birkhäuser Boston, Boston, MA, 1983, p. 1-42 | DOI | MR | Zbl

[10] G. Eskin - “On non-abelian Radon transform”, Russian J. Math. Phys. 11 (2004) no. 4, p. 391-408 | MR | Zbl

[11] G. Eskin & J. Ralston - “On the inverse boundary value problem for linear isotropic elasticity and Cauchy-Riemann systems”, in Inverse problems and spectral theory, Contemp. Math., vol. 348, American Mathematical Society, Providence, RI, 2004, p. 53-69 | DOI | MR | Zbl

[12] D. Finch & G. Uhlmann - “The X-ray transform for a non-Abelian connection in two dimensions”, Inverse Problems 17 (2001) no. 4, p. 695-701 | DOI | MR | Zbl

[13] F. Forstnerič - Stein manifolds and holomorphic mappings, Ergeb. Math. Grenzgeb. (3), vol. 56, Springer, Heidelberg, 2011 | DOI

[14] H. Grauert - “Analytische Faserungen über holomorph-vollständigen Räumen”, Math. Ann. 135 (1958), p. 263-273 | DOI | Zbl

[15] V. Guillemin & D. Kazhdan - “Some inverse spectral results for negatively curved $2$ -manifolds”, Topology 19 (1980) no. 3, p. 301-312 | DOI | MR | Zbl

[16] R. S. Hamilton - “The inverse function theorem of Nash and Moser”, Bull. Amer. Math. Soc. (N.S.) 7 (1982) no. 1, p. 65-222 | DOI | MR | Zbl

[17] A. Hilger, I. Manke, N. Kardjilov et al. - “Tensorial neutron tomography of three-dimensional magnetic vector fields in bulk materials”, Nat Commun 9 (2018), article ID 4023, 7 pages | DOI

[18] N. J. Hitchin - “Complex manifolds and Einstein’s equations”, in Twistor geometry and nonlinear systems (Primorsko, 1980), Lect. Notes in Math., vol. 970, Springer, Berlin-New York, 1982, p. 73-99 | DOI | MR | Zbl

[19] C. LeBrun - Spaces of complex geodesics and related structures, Ph. D. Thesis, University of Oxford, 1980

[20] C. LeBrun & L. J. Mason - “Zoll manifolds and complex surfaces”, J. Differential Geom. 61 (2002) no. 3, p. 453-535 | MR | Zbl

[21] C. LeBrun & L. J. Mason - “Zoll metrics, branched covers, and holomorphic disks”, Comm. Anal. Geom. 18 (2010) no. 3, p. 475-502 | DOI | MR | Zbl

[22] J. Leiterer - “Holomorphic vector bundles and the Oka-Grauert principle”, in Several complex variables. IV. Algebraic aspects of complex analysis, Encycl. Math. Sci., vol. 10, Springer, Berlin, 1990, p. 63-103 | DOI

[23] B. Malgrange - Lectures on the theory of functions of several complex variables, T.I.F.R. Lectures on Math. and Physics, vol. 13, Tata Institute of Fundamental Research, Bombay, 1984

[24] L. J. Mason - “Global anti-self-dual Yang-Mills fields in split signature and their scattering”, J. reine angew. Math. 597 (2006), p. 105-133 | DOI | MR | Zbl

[25] L. J. Mason & N. M. J. Woodhouse - Integrability, self-duality, and twistor theory, London Math. Society Monographs. New Series, vol. 15, The Clarendon Press, Oxford University Press, New York, 1996

[26] T. Mettler - “Metrisability of projective surfaces and pseudo-holomorphic curves”, Math. Z. 298 (2021) no. 1-2, p. 69-78 | DOI | MR | Zbl

[27] T. Mettler & G. P. Paternain - “Convex projective surfaces with compatible Weyl connection are hyperbolic”, Anal. PDE 13 (2020) no. 4, p. 1073-1097 | DOI | MR | Zbl

[28] F. Monard, R. Nickl & G. P. Paternain - “Consistent inversion of noisy non-Abelian X-ray transforms”, Comm. Pure Appl. Math. 74 (2021) no. 5, p. 1045-1099 | DOI | MR | Zbl

[29] F. Monard, R. Nickl & G. P. Paternain - “Statistical guarantees for Bayesian uncertainty quantification in nonlinear inverse problems with Gaussian process priors”, Ann. Statist. 49 (2021) no. 6, p. 3255-3298 | DOI | MR | Zbl

[30] R. G. Novikov - “On determination of a gauge field on $ℝ^{d}$ from its non-abelian Radon transform along oriented straight lines”, J. Inst. Math. Jussieu 1 (2002) no. 4, p. 559-629 | DOI | MR | Zbl

[31] R. G. Novikov - “Non-Abelian Radon transform and its applications”, in The Radon transform—the first 100 years and beyond, Radon Ser. Comput. Appl. Math., vol. 22, Walter de Gruyter, Berlin, 2019, p. 115-127 | DOI | MR | Zbl

[32] N. R. O’Brian & J. H. Rawnsley - “Twistor spaces”, Ann. Global Anal. Geom. 3 (1985) no. 1, p. 29-58 | DOI | MR | Zbl

[33] G. P. Paternain & M. Salo - “The non-Abelian X-ray transform on surfaces”, 2020, to appear in J. Differential Geom. | arXiv

[34] G. P. Paternain, M. Salo & G. Uhlmann - “The attenuated ray transform for connections and Higgs fields”, Geom. Funct. Anal. 22 (2012) no. 5, p. 1460-1489 | DOI | MR | Zbl

[35] G. P. Paternain, M. Salo & G. Uhlmann - “On the range of the attenuated ray transform for unitary connections”, Internat. Math. Res. Notices (2015) no. 4, p. 873-897 | DOI | MR | Zbl

[36] G. P. Paternain, M. Salo & G. Uhlmann - Geometric inverse problems—with emphasis on two dimensions, Cambridge Studies in Advanced Math., vol. 204, Cambridge University Press, Cambridge, 2023 | DOI

[37] G. P. Paternain, M. Salo, G. Uhlmann & H. Zhou - “The geodesic X-ray transform with matrix weights”, Amer. J. Math. 141 (2019) no. 6, p. 1707-1750 | DOI | MR | Zbl

[38] R. Penrose - “The twistor programme”, Rep. Mathematical Phys. 12 (1977) no. 1, p. 65-76 | DOI | MR

[39] L. Pestov & G. Uhlmann - “On characterization of the range and inversion formulas for the geodesic X-ray transform”, Internat. Math. Res. Notices (2004) no. 80, p. 4331-4347 | DOI | MR | Zbl

[40] L. Pestov & G. Uhlmann - “Two dimensional compact simple Riemannian manifolds are boundary distance rigid”, Ann. of Math. (2) 161 (2005) no. 2, p. 1093-1110 | DOI | MR | Zbl

[41] A. Pressley & G. Segal - Loop groups, Oxford Math. Monographs, The Clarendon Press, Oxford University Press, New York, 1986

[42] M. Sales, M. Strobl, T. Shinohara et al. - “Three dimensional polarimetric neutron tomography of magnetic fields”, Nat Sci Rep 8 (2018), article ID 2214, 6 pages | DOI

[43] M. Salo & G. Uhlmann - “The attenuated ray transform on simple surfaces”, J. Differential Geom. 88 (2011) no. 1, p. 161-187 | MR | Zbl

[44] R. T. Seeley - “Extension of $C^{\infty}$ functions defined in a half space”, Proc. Amer. Math. Soc. 15 (1964), p. 625-626 | DOI | MR | Zbl

[45] V. A. Sharafutdinov - Integral geometry of tensor fields, Inverse and Ill-posed Problems Series, VSP, Utrecht, 1994 | DOI

[46] V. A. Sharafutdinov - “On the inverse problem of determining a connection on a vector bundle”, J. Inverse Ill-Posed Probl. 8 (2000) no. 1, p. 51-88 | DOI | MR | Zbl

[47] I. M. Singer & J. A. Thorpe - Lecture notes on elementary topology and geometry, Undergraduate Texts in Math., Springer-Verlag, New York-Heidelberg, 1976 | DOI

[48] P. Stefanov, G. Uhlmann & A. Vasy - “Local and global boundary rigidity and the geodesic X-ray transform in the normal gauge”, Ann. of Math. (2) 194 (2021) no. 1, p. 1-95 | DOI | MR | Zbl

[49] G. Uhlmann & A. Vasy - “The inverse problem for the local geodesic ray transform”, Invent. Math. 205 (2016) no. 1, p. 83-120 | DOI | MR | Zbl

[50] L. B. Vertgeim - “Integral geometry with a matrix weight and a nonlinear problem of the reconstruction of matrices”, Dokl. Akad. Nauk SSSR 319 (1991) no. 3, p. 531-534

Cité par Sources :