Parabolic bundles and spherical metrics
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- by Martin de Borbon and Dmitri Panov PDF
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Abstract:
We use the Kobayashi-Hitchin correspondence for parabolic bundles to reprove the results of Troyanov [Trans. Amer. Math. Soc. 324 (1991), pp. 793-821] and Luo-Tian [Proc. Amer. Math. Soc. 116 (1992), pp. 1119-1129] regarding existence and uniqueness of conformal spherical metrics on the Riemann sphere with prescribed cone angles in the interval $(0, 2\pi )$ at a given configuration of three or more points.References
- A. Borel, P.-P. Grivel, B. Kaup, A. Haefliger, B. Malgrange, and F. Ehlers, Algebraic $D$-modules, Perspectives in Mathematics, vol. 2, Academic Press, Inc., Boston, MA, 1987. MR 882000
- Daniele Bartolucci, Changfeng Gui, Aleks Jevnikar, and Amir Moradifam, A singular sphere covering inequality: uniqueness and symmetry of solutions to singular Liouville-type equations, Math. Ann. 374 (2019), no. 3-4, 1883–1922. MR 3985126, DOI 10.1007/s00208-018-1761-1
- Olivier Biquard, Fibrés paraboliques stables et connexions singulières plates, Bull. Soc. Math. France 119 (1991), no. 2, 231–257 (French, with English summary). MR 1116847, DOI 10.24033/bsmf.2166
- Indranil Biswas, A criterion for the existence of a flat connection on a parabolic vector bundle, Adv. Geom. 2 (2002), no. 3, 231–241. MR 1924757, DOI 10.1515/advg.2002.011
- Marco Brunella, Birational geometry of foliations, IMPA Monographs, vol. 1, Springer, Cham, 2015. MR 3328860, DOI 10.1007/978-3-319-14310-1
- Pierre Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, Vol. 163, Springer-Verlag, Berlin-New York, 1970 (French). MR 0417174, DOI 10.1007/BFb0061194
- Alexandre Eremenko, Metrics of constant positive curvature with conic singularities. A survey. arXiv:2103.13364, 2021.
- Yulij Ilyashenko and Sergei Yakovenko, Lectures on analytic differential equations, Graduate Studies in Mathematics, vol. 86, American Mathematical Society, Providence, RI, 2008. MR 2363178, DOI 10.1090/gsm/086
- Semin Kim and Graeme Wilkin, Analytic convergence of harmonic metrics for parabolic Higgs bundles, J. Geom. Phys. 127 (2018), 55–67. MR 3774336, DOI 10.1016/j.geomphys.2018.01.023
- Frank Loray and David Marín Pérez, Projective structures and projective bundles over compact Riemann surfaces, Astérisque 323 (2009), 223–252 (English, with English and French summaries). MR 2647972
- Frank Loray, Masa-Hiko Saito, and Carlos Simpson, Foliations on the moduli space of rank two connections on the projective line minus four points, Geometric and differential Galois theories, Sémin. Congr., vol. 27, Soc. Math. France, Paris, 2013, pp. 117–170 (English, with English and French summaries). MR 3203552
- Lingguang Li, Jijian Song, and Bin Xu, Irreducible cone spherical metrics and stable extensions of two line bundles, Adv. Math. 388 (2021), Paper No. 107854, 36. MR 4283757, DOI 10.1016/j.aim.2021.107854
- Feng Luo and Gang Tian, Liouville equation and spherical convex polytopes, Proc. Amer. Math. Soc. 116 (1992), no. 4, 1119–1129. MR 1137227, DOI 10.1090/S0002-9939-1992-1137227-5
- Takuro Mochizuki, Kobayashi-Hitchin correspondence for tame harmonic bundles and an application, Astérisque 309 (2006), viii+117 (English, with English and French summaries). MR 2310103
- Gabriele Mondello and Dmitri Panov, Spherical metrics with conical singularities on a 2-sphere: angle constraints, Int. Math. Res. Not. IMRN 16 (2016), 4937–4995. MR 3556430, DOI 10.1093/imrn/rnv300
- V. B. Mehta and C. S. Seshadri, Moduli of vector bundles on curves with parabolic structures, Math. Ann. 248 (1980), no. 3, 205–239. MR 575939, DOI 10.1007/BF01420526
- Dmitri Panov, Polyhedral Kähler manifolds, Geom. Topol. 13 (2009), no. 4, 2205–2252. MR 2507118, DOI 10.2140/gt.2009.13.2205
- Yanir A. Rubinstein, Smooth and singular Kähler-Einstein metrics, Geometric and spectral analysis, Contemp. Math., vol. 630, Amer. Math. Soc., Providence, RI, 2014, pp. 45–138. MR 3328541, DOI 10.1090/conm/630/12665
- Carlos T. Simpson, Harmonic bundles on noncompact curves, J. Amer. Math. Soc. 3 (1990), no. 3, 713–770. MR 1040197, DOI 10.1090/S0894-0347-1990-1040197-8
- Dennis Sullivan and William Thurston, Manifolds with canonical coordinate charts: some examples, Enseign. Math. (2) 29 (1983), no. 1-2, 15–25. MR 702731
- Marc Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 (1991), no. 2, 793–821. MR 1005085, DOI 10.1090/S0002-9947-1991-1005085-9
Additional Information
- Martin de Borbon
- Affiliation: Department of Mathematics, King’s College London, Strand, London, WC2R 2LS, United Kingdom
- MR Author ID: 1204130
- ORCID: 0000-0002-9078-4624
- Email: martin.deborbon@kcl.ac.uk
- Dmitri Panov
- Affiliation: Department of Mathematics, King’s College London, Strand, London, WC2R 2LS, United Kingdom
- MR Author ID: 606559
- Email: dmitri.panov@kcl.ac.uk
- Received by editor(s): September 27, 2021
- Received by editor(s) in revised form: February 18, 2022
- Published electronically: September 23, 2022
- Additional Notes: This work was supported by EPSRC Project EP/S035788/1, Kähler manifolds of constant curvature with conical singularities.
- Communicated by: Jiaping Wang
- © Copyright 2022 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 150 (2022), 5459-5472
- MSC (2020): Primary 57M50, 32L05; Secondary 34M35, 53C45, 32S65
- DOI: https://doi.org/10.1090/proc/16052