Skip to main content
Springer Nature Link
Log in

Riemannian Surfaces with Simple Singularities

  • Chapter
  • First Online:
Reshetnyak's Theory of Subharmonic Metrics

  • 161 Accesses

Abstract

In this note we discuss the geometry of Riemannian surfaces having a discrete set of singular points. We assume the conformal structure extends through the singularities and the curvature is integrable. Such points are called simple singularities. We first describe them locally and then globally using the notion of (real) divisor. We formulate a Gauss–Bonnet formula and relate it to some asymptotic isoperimetric ratio. We prove a classifications theorem for flat metrics with simple singularities on a compact surface and discuss the Berger–Nirenberg Problem on surfaces with a divisor. We finally discuss the relation with spherical polyhedra.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
eBook
USD 109.00
Price excludes VAT (USA)
Softcover Book
USD 139.99
Price excludes VAT (USA)
Hardcover Book
USD 139.99
Price excludes VAT (USA)

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Alexandrov, A.D.: A. D. Alexandrov selected works. Part II. Chapman & Hall/CRC, Boca Raton, FL (2006). Intrinsic geometry of convex surfaces, Edited by S. S. Kutateladze, Translated from the Russian by S. Vakhrameyev

    Google Scholar 

  2. Finn, R.: On a class of conformal metrics, with application to differential geometry in the large. Comment. Math. Helv. 40, 1–30 (1965). https://doi.org/10.1007/BF02564362

    Article  MathSciNet  MATH  Google Scholar 

  3. Hulin, D.: Déformation conforme des surfaces non compactes à courbure négative. J. Funct. Anal. 111(2), 449–472 (1993). https://doi.org/10.1006/jfan.1993.1021

    Article  MathSciNet  MATH  Google Scholar 

  4. Hulin, D., Troyanov, M.: Sur la courbure des surfaces de Riemann ouvertes. C. R. Acad. Sci. Paris Sér. I Math. 310(4), 203–206 (1990)

    MathSciNet  MATH  Google Scholar 

  5. Hulin, D., Troyanov, M.: Prescribing curvature on open surfaces. Math. Ann. 293(2), 277–315 (1992). https://doi.org/10.1007/BF01444716

    Article  MathSciNet  MATH  Google Scholar 

  6. Kazdan, J.L., Warner, F.W.: Curvature functions for compact 2-manifolds. Ann. of Math. (2) 99, 14–47 (1974). https://doi.org/10.2307/1971012

  7. Lai, M.: Metric aspects of conic surfaces. Front. Math. China 11(5), 1291–1312 (2016). https://doi.org/10.1007/s11464-016-0556-5

    Article  MathSciNet  MATH  Google Scholar 

  8. Luo, F., Tian, G.: Liouville equation and spherical convex polytopes. Proc. Amer. Math. Soc. 116(4), 1119–1129 (1992). https://doi.org/10.2307/2159498

    Article  MathSciNet  MATH  Google Scholar 

  9. McOwen, R.C.: Prescribed curvature and singularities of conformal metrics on Riemann surfaces. J. Math. Anal. Appl. 177(1), 287–298 (1993). https://doi.org/10.1006/jmaa.1993.1258

    Article  MathSciNet  MATH  Google Scholar 

  10. Picard, É.: De l’intégration de l’équation Δu = eu sur une surface de Riemann fermée. J. Reine Angew. Math. 130, 243–258 (1905). https://doi.org/10.1515/crll.1905.130.243

    Article  MathSciNet  Google Scholar 

  11. Shiohama, K.: Total curvatures and minimal areas of complete surfaces. Proc. Amer. Math. Soc. 94(2), 310–316 (1985). https://doi.org/10.2307/2045396

    MathSciNet  MATH  Google Scholar 

  12. Tang, J.J.: Prescribing curvature with negative total curvature on open Riemann surfaces. Proc. Amer. Math. Soc. 116(4), 1023–1030 (1992). https://doi.org/10.2307/2159484

    Article  MathSciNet  MATH  Google Scholar 

  13. Troyanov, M.: Les surfaces euclidiennes à singularités coniques. Enseign. Math. (2) 32(1–2), 79–94 (1986)

    Google Scholar 

  14. Troyanov, M.: Coordonnées polaires sur les surfaces riemanniennes singulières. Ann. Inst. Fourier (Grenoble) 40(4), 913–937 (1991) (1990). https://doi.org/10.5802/aif.1241

  15. Troyanov, M.: Prescribing curvature on compact surfaces with conical singularities. Trans. Amer. Math. Soc. 324(2), 793–821 (1991). https://doi.org/10.2307/2001742

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

First published as: Surfaces riemanniennes à singularités simples. In Differential geometry: geometry in mathematical physics and related topics, 619–628, Proc. Sympos. Pure Math., 54, Part 2, Amer. Math. Soc., Providence, RI, 1993. Translated from French by the author.

Author information

Authors and Affiliations

  1. Section de Mathématiques, École Polytechnique Féderale de Lausanne, 1015, Lausanne, Switzerland

    Marc Troyanov

Authors
  1. Marc Troyanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marc Troyanov .

Editor information

Editors and Affiliations

  1. IMAG, Université de Montpellier, Montpellier, France

    François Fillastre

  2. ENGIE Lab CRIGEN, Pierrefitte-Stains, France

    Dmitriy Slutskiy

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Troyanov, M. (2023). Riemannian Surfaces with Simple Singularities. In: Fillastre, F., Slutskiy, D. (eds) Reshetnyak's Theory of Subharmonic Metrics. Springer, Cham. https://doi.org/10.1007/978-3-031-24255-7_3

Download citation

Publish with us

Policies and ethics

Search

Navigation