Skip to main content
SpringerLink
Log in

Trivalent Maximal Surfaces in Minkowski Space

  • Conference paper
  • First Online:
Lorentzian Geometry and Related Topics (GELOMA 2016)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 211))

Included in the following conference series:

  • 567 Accesses

Abstract

We investigate discretizations of maximal surfaces in Minkowski space, which are surfaces with vanishing mean curvature. The corresponding discrete surfaces admit a Weierstrass-type representation in terms of discrete holomorphic quadratic differentials. There are two particular types of discrete maximal surfaces that are obtained by taking the real part and the imaginary part of the representation formula, and they are deformable to each other by a one-parameter family. We further introduce a compatible notion of vertex normals for general trivalent surfaces to characterize their singularities in Minkowski space as in the smooth theory.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. A.I. Bobenko, U. Pinkall, Discrete surfaces with constant negative Gaussian curvature and the Hirota equation. J. Differential Geom. 43(3), 527–611 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  2. A.I. Bobenko, U. Pinkall, Discrete isothermic surfaces. J. Reine Angew. Math. 475, 187–208 (1996)

    MathSciNet  MATH  Google Scholar 

  3. A.I. Bobenko, H. Pottmann, J. Wallner, A curvature theory for discrete surfaces based on mesh parallelity. Math. Ann. 348, 1–24 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. A.I. Bobenko, W.K. Schief, Affine spheres: discretization via duality relations. Experiment. Math. 8(3), 261–280 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  5. A.I. Bobenko, Y. Suris, Discrete differential geometry: Integrable structure, Graduate Studies in Mathematics, 98 (American Mathematical Society, Providence, RI, 2008)

    Book  Google Scholar 

  6. E. Calabi, Examples of Bernstein problems for some non-linear equations. Proc. Sympos. Pure Math. 15, 223–230 (1970)

    Article  Google Scholar 

  7. S. Fujimori, K. Saji, M. Umehara, K. Yamada, Singularities of maximal surfaces. Math. Z. 259, 827–848 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. U. Hertrich-Jeromin, Transformations of discrete isothermic nets and discrete cmc-1 surfaces in hyperbolic space. Manuscripta Math. 102(4), 465–486 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. U. Hertrich-Jeromin, T. Hoffmann, U. Pinkall, A discrete version of the Darboux transform for isothermic surfaces, Discrete integrable geometry and physics (Vienna, 1996), 59–81, Oxford Lecture Ser. Math. Appl., 16, Oxford Univ. Press, New York (1999)

    Google Scholar 

  10. R. Hirota, Nonlinear partial difference equations. III. Discrete sine-Gordon equation. J. Phys. Soc. Japan 43(6), 2079–2086 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  11. O. Kobayashi, Maximal surfaces in the \(3\)-dimensional Minkowski space \(\mathbb{L}^{3}\). Tokyo J. Math. 6, 297–309 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  12. O. Kobayashi, Maximal surfaces with conelike singularities. J. Math. Soc. Japan 36(4), 609–617 (1984)

    Google Scholar 

  13. W.Y. Lam, Discrete minimal surfaces: critical points of the area functional from integrable systems, to appear in IMRN, available from arXiv:1510.08788

  14. W.Y. Lam, U. Pinkall, Isothermic triangulated surfaces, Math. Ann. 368, no. 1–2, 165–195 (2017)

    Google Scholar 

  15. W.Y. Lam, U. Pinkall, Holomorphic vector fields and quadratic differentials on planar triangular meshes, Adv. discrete Differ. Geom. 241–265, Springer, [Berlin] (2016)

    Google Scholar 

  16. N. Matsuura, H. Urakawa, Discrete improper affine spheres. J. Geom. Phys. 45(1–2), 164–183 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Y. Ogata, M. Yasumoto, Construction of discrete constant mean curvature surfaces in Riemannian spaceforms and applications, Differ. Geom. Appl. 54, part A, 264–281 (2017)

    Google Scholar 

  18. U. Pinkall, K. Polthier, Computing discrete minimal surfaces and their conjugates. Experiment. Math. 2(1), 15–36 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  19. K. Polthier, W. Rossman, Discrete constant mean curvature surfaces and their index. J. Reine Angew. Math. 549, 47–77 (2002)

    MathSciNet  MATH  Google Scholar 

  20. W. Rossman, M. Yasumoto, Discrete linear Weingarten surfaces and their singularities in Riemannian and Lorentzian spaceforms, to appear in Advanced Studies in Pure Mathematics

    Google Scholar 

  21. M. Umehara, K. Yamada, Maximal surfaces with singularities in Mikowski space. Hokkaido Math. J. 35, 13–40 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. M. Yasumoto, Discrete maximal surfaces with singularities in Minkowski space. Differential Geom. Appl. 43, 130–154 (2015)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The second author was supported by the JSPS Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers “Mathematical Science of Symmetry, Topology and Moduli, Evolution of International Research Network based on OCAMI”.

Author information

Authors and Affiliations

  1. Department of Mathematics, Brown University, Box 1917, Providence, RI, 02912, USA

    Wai Yeung Lam

  2. Osaka City University Advanced Mathematical Institute, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan

    Masashi Yasumoto

Authors
  1. Wai Yeung Lam
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Masashi Yasumoto
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Masashi Yasumoto .

Editor information

Editors and Affiliations

  1. Department of Algebra, Geometry, Topology, University of Malaga, Málaga, Spain

    María A. Cañadas-Pinedo

  2. Department of Algebra, Geometry, Topology, University of Malaga, Málaga, Spain

    José Luis Flores

  3. Department of Applied Mathematics, University of Malaga, Malaga, Spain

    Francisco J. Palomo

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Lam, W.Y., Yasumoto, M. (2017). Trivalent Maximal Surfaces in Minkowski Space . In: Cañadas-Pinedo, M., Flores, J., Palomo, F. (eds) Lorentzian Geometry and Related Topics. GELOMA 2016. Springer Proceedings in Mathematics & Statistics, vol 211. Springer, Cham. https://doi.org/10.1007/978-3-319-66290-9_10

Download citation

Publish with us

Policies and ethics

Search

Navigation