Skip to main content
SpringerLink
Log in

Reduction of Vector Boundary Value Problems on Riemann Surfaces to One-Dimensional Problems

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

This article lays foundations for the theory of vector conjugation boundary value problems on a compact Riemann surface of arbitrary positive genus. The main constructions of the classical theory of vector boundary value problems on the plane are carried over to Riemann surfaces: reduction of the problem to a system of integral equations on a contour, the concepts of companion and adjoint problems, as well as their connection with the original problem, the construction of a meromorphic matrix solution. We show that each vector conjugation boundary value problem reduces to a problem with a triangular coefficient matrix, which in fact reduces the problem to a succession of one-dimensional problems. This reduction to the well-understood one-dimensional problems opens up a path towards a complete construction of the general solution of vector boundary value problems on Riemann surfaces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Semenko E. V., “Connection between holomorphic vector bundles and cohomology on a Riemann surface and conjugation boundary value problems,” Sib. Math. J., vol. 58, No. 2, 310–318 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  2. Vekua N. P., Systems of Singular Integral Equations and Some Boundary Value Problems [Russian], Gostekhizdat, Moscow (1950).

    Google Scholar 

  3. Grothendieck A., “Sur la classification des fibres holomorphes sur la sphere de Riemann,” Amer. J. Math., vol. 79, No. 1, 121–138 (1957).

    Article  MathSciNet  MATH  Google Scholar 

  4. Gunning R. C., Lectures on Vector Bundles over Riemann Surfaces, Princeton Univ. Press, Princeton (1967).

    MATH  Google Scholar 

  5. Tyurin A. N., “On the classification of two–dimensional fibre bundles over an algebraic curve of arbitrary genus,” Izv. Akad. Nauk SSSR Ser. Mat., vol. 28, No. 1, 21–52 (1964).

    MathSciNet  Google Scholar 

  6. Tyurin A. N., “The classification of vector bundles over an algebraic curve of arbitrary genus,” Amer. Math. Soc. Transl. Ser. 2, vol. 63, No. 3, 245–279 (1967,).

    MATH  Google Scholar 

  7. Tyurin A. N., “Classification of n–dimensional vector bundles over an algebraic curve of arbitrary genus,” Amer. Math. Soc. Transl. Ser. 2, vol. 73, No. 6, 196–211 (1968).

    MATH  Google Scholar 

  8. Zverovich E. I., “Boundary value problems in the theory of analytic functions in Hölder classes on Riemann surfaces,” Russian Math. Surveys, vol. 26, No. 1, 117–192 (1971).

    Article  MATH  Google Scholar 

  9. Monakhov V. N., Verkhovod D. B., and Semenko E. V., Boundary–Value Problems for Quasianalytic Functions on Compact Riemann Surfaces [Russian], Lavrentev Inst. of Hydrodynamics, Novosibirsk (1990).

    MATH  Google Scholar 

  10. Monakhov V. N. and Semenko E. V., Boundary–Value Problems and Pseudodifferential Operators on Riemann Surfaces [Russian], Fizmatlit, Moscow (2003).

    MATH  Google Scholar 

  11. Semenko E. V., “Prym differentials as solutions to boundary value problems on Riemann surfaces,” Sib. Math. J., vol. 57, No. 1, 124–134 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  12. Forster O., Lectures on Riemann Surfaces, Springer–Verlag, New York (1981).

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Novosibirsk State Technical University Novosibirsk State Pedagogical University, Novosibirsk, Russia

    E. V. Semenko

Authors
  1. E. V. Semenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. V. Semenko.

Additional information

__________

Translated from Sibirskii Matematicheskii Zhurnal, vol. 60, no. 1, pp. 201–213, January–February, 2019; DOI: 10.17377/smzh.2019.60.117.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Semenko, E.V. Reduction of Vector Boundary Value Problems on Riemann Surfaces to One-Dimensional Problems. Sib Math J 60, 153–163 (2019). https://doi.org/10.1134/S0037446619010178

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0037446619010178

Keywords

Search

Navigation