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Mathematical Notes

, Volume 77, Issue 5–6, pp 595–605 | Cite as

Constructive Solvability Conditions for the Riemann-Hilbert Problem

  • I. V. V’yugin
Article

Abstract

Sufficient and necessary conditions for the solvability of the Riemann-Hilbert problem are studied. These conditions consist in the possibility of constructing stable and semistable pairs (of bundles and connections) for a given monodromy. The obtained results make it possible to develop algorithms for testing the solvability conditions for the Riemann-Hilbert problem.

Key words

Riemann-Hilbert problem Fuchsian system monodromy representation bundle valuation stable pair of a bundle and a logarithmic connection semistable pair of a bundle and a logarithmic connection 

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REFERENCES

  1. 1.
    A. A. Bolibrukh, “Hilbert’s 21st problem for Fuchsian linear systems,” Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.], 206 (1994).Google Scholar
  2. 2.
    V. P. Kostov, “Fuchsian systems on ℂP 1 and the Riemann-Hilbert Problem,” C. R. Acad. Sci. Paris Ser. I, 315 (1992), 143–148.Google Scholar
  3. 3.
    A. A. Bolibrukh, Fuchsian Differential Equations and Holomorphic Bundles [in Russian], MTsNMO, Moscow, 2000.Google Scholar
  4. 4.
    A. A. Bolibrukh, “The Riemann-Hilbert problem on a compact Riemann surface,” Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.], 238 (2002), 55–69.Google Scholar
  5. 5.
    A. I. Gladyshev, “On the Riemann-Hilbert problem in dimension 4,” J. Dynamical Control Systems, 6 (2000), no. 2, 219–264.CrossRefGoogle Scholar
  6. 6.
    S. Malek, “Systemes fuchsiens a monodromie reductible,” C. R. Acad. Sci. Paris Ser. I, 332 (2001), no. 8, 691–694.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • I. V. V’yugin
    • 1
  1. 1.M. V. Lomonosov Moscow State UniversityMoscowRussia

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