Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A Riemannian four vertex theorem for surfaces with boundary


Author: Mohammad Ghomi
Journal: Proc. Amer. Math. Soc. 139 (2011), 293-303
MSC (2010): Primary 53C20, 53C22; Secondary 53A04, 53A05
DOI: https://doi.org/10.1090/S0002-9939-2010-10507-5
Published electronically: July 22, 2010
MathSciNet review: 2729091
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that every metric of constant curvature on a compact surface $ M$ with boundary $ \partial M$ induces at least four vertices, i.e., local extrema of geodesic curvature on a connected component of $ \partial M$ if, and only if, $ M$ is simply connected. Indeed, when $ M$ is not simply connected, we construct hyperbolic, parabolic, and elliptic metrics of constant curvature on $ M$ with only two critical points of geodesic curvature on each component of $ \partial M$. With few exceptions, these metrics are obtained by removing the singularities and a perturbation of flat structures on closed surfaces.


References [Enhancements On Off] (What's this?)

  • 1. R. Benedetti and C. Petronio.
    Lectures on hyperbolic geometry.
    Universitext. Springer-Verlag, Berlin, 1992. MR 1219310 (94e:57015)
  • 2. S. Brendle.
    Curvature flows on surfaces with boundary.
    Math. Ann., 324(3):491-519, 2002. MR 1938456 (2003j:53103)
  • 3. G. Cairns, M. Özdemir, and E.-H. Tjaden.
    A counterexample to a conjecture of U. Pinkall.
    Topology, 31(3):557-558, 1992. MR 1174258 (93d:53004)
  • 4. S. I. R. Costa and M. Firer.
    Four-or-more-vertex theorems for constant curvature manifolds.
    In Real and complex singularities (São Carlos, 1998), volume 412 of Chapman & Hall/CRC Res. Notes Math., pages 164-172. Chapman & Hall/CRC, Boca Raton, FL, 2000. MR 1715702 (2001d:51013)
  • 5. D. DeTurck, H. Gluck, D. Pomerleano, and D. S. Vick.
    The four vertex theorem and its converse.
    Notices Amer. Math. Soc., 54(2):192-207, 2007. MR 2285124 (2007k:53003)
  • 6. M. Ghomi.
    Vertices of closed curves in Riemannian surfaces.
    Preprint available at www.math.gatech.edu/$ \sim$ghomi. arXiv:1006.4182v1
  • 7. M. Ghomi and M. Kossowski.
    $ h$-principles for hypersurfaces with prescribed principal curvatures and directions.
    Trans. Amer. Math. Soc., 358(10):4379-4393 (electronic), 2006. MR 2231382 (2007c:53072)
  • 8. H. Gluck.
    The generalized Minkowski problem in differential geometry in the large.
    Ann. of Math. (2), 96:245-276, 1972. MR 0309021 (46:8132)
  • 9. B. Guan and J. Spruck.
    Locally convex hypersurfaces of constant curvature with boundary.
    Comm. Pure Appl. Math., 57(10):1311-1331, 2004. MR 2069725 (2005d:53097)
  • 10. M. W. Hirsch.
    Differential topology.
    Springer-Verlag, New York, 1994.
    Corrected reprint of the 1976 original. MR 1336822 (96c:57001)
  • 11. S. B. Jackson.
    Vertices for plane curves.
    Bull. Amer. Math. Soc., 50:564-478, 1944. MR 0010992 (6:100e)
  • 12. S. B. Jackson.
    The four-vertex theorem for surfaces of constant curvature.
    Amer. J. Math., 67:563-582, 1945. MR 0014257 (7:259h)
  • 13. A. Kneser.
    Bemerkungen über die anzahl der extrema des krümmung auf geschlossenen kurven und über verwandte fragen in einer night eucklidischen geometrie.
    In Festschrift Heinrich Weber, pages 170-180. Teubner, 1912.
  • 14. M. Maeda.
    The four-or-more vertex theorems in $ 2$-dimensional space forms.
    Nat. Sci. J. Fac. Educ. Hum. Sci. Yokohama Natl. Univ., (1):43-46, 1998. MR 1710269 (2001a:53023)
  • 15. H. Masur and S. Tabachnikov.
    Rational billiards and flat structures.
    In Handbook of dynamical systems, Vol. 1A, pages 1015-1089. North-Holland, Amsterdam, 2002. MR 1928530 (2003j:37002)
  • 16. S. Mukhopadhyaya.
    New methods in the geometry of a plane arc.
    Bull. Calcutta Math. Soc. I, pages 31-37, 1909.
  • 17. B. Osgood, R. Phillips, and P. Sarnak.
    Extremals of determinants of Laplacians.
    J. Funct. Anal., 80(1):148-211, 1988. MR 960228 (90d:58159)
  • 18. V. Ovsienko and S. Tabachnikov.
    Projective differential geometry old and new, volume 165 of Cambridge Tracts in Mathematics.
    Cambridge University Press, Cambridge, 2005.
    From the Schwarzian derivative to the cohomology of diffeomorphism groups. MR 2177471 (2007b:53017)
  • 19. U. Pinkall.
    On the four-vertex theorem.
    Aequationes Math., 34(2-3):221-230, 1987. MR 921101 (89a:53007)
  • 20. A. V. Pogorelov.
    Extrinsic geometry of convex surfaces.
    Translated from the Russian by Israel Program for Scientific Translations, Translations of Mathematical Monographs, Vol. 35.
    American Mathematical Society, Providence, RI, 1973. MR 0346714 (49:11439)
  • 21. J. G. Ratcliffe.
    Foundations of hyperbolic manifolds, volume 149 of Graduate Texts in Mathematics.
    Springer, New York, second edition, 2006. MR 2249478 (2007d:57029)
  • 22. E. Raphaël, J.-M. di Meglio, M. Berger and E. Calabi.
    Convex particles at interfaces, J. Phys. I France 2:571-579, 1992.
  • 23. M. Spivak.
    A comprehensive introduction to differential geometry. Vol. II.
    Publish or Perish Inc., Wilmington, Del., second edition, 1979. MR 0532831 (82g:53003b)
  • 24. G. Thorbergsson and M. Umehara.
    A unified approach to the four vertex theorems. II.
    In Differential and symplectic topology of knots and curves, volume 190 of Amer. Math. Soc. Transl. Ser. 2, pages 229-252. Amer. Math. Soc., Providence, RI, 1999. MR 1738398 (2001f:53009)
  • 25. W. P. Thurston.
    Shapes of polyhedra and triangulations of the sphere.
    In The Epstein birthday schrift, volume 1 of Geom. Topol. Monogr., pages 511-549 (electronic). Geom. Topol. Publ., Coventry, 1998. MR 1668340 (2000b:57026)
  • 26. M. Troyanov.
    Les surfaces euclidiennes à singularités coniques.
    Enseign. Math. (2), 32(1-2): 79-94, 1986. MR 850552 (87i:30079)
  • 27. M. Troyanov.
    On the moduli space of singular Euclidean surfaces.
    In Handbook of Teichmüller theory. Vol. I, volume 11 of IRMA Lect. Math. Theor. Phys., pages 507-540. Eur. Math. Soc., Zürich, 2007. MR 2349679 (2009b:57041)
  • 28. N. S. Trudinger and X.-J. Wang.
    On locally convex hypersurfaces with boundary.
    J. Reine Angew. Math., 551:11-32, 2002. MR 1932171 (2004b:58017)
  • 29. M. Umehara.
    $ 6$-vertex theorem for closed planar curve which bounds an immersed surface with nonzero genus.
    Nagoya Math. J., 134:75-89, 1994. MR 1280654 (95e:53007)
  • 30. M. Umehara.
    A unified approach to the four vertex theorems. I.
    In Differential and symplectic topology of knots and curves, volume 190 of Amer. Math. Soc. Transl. Ser. 2, pages 185-228. Amer. Math. Soc., Providence, RI, 1999. MR 1738397 (2001f:53008)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 53C20, 53C22, 53A04, 53A05

Retrieve articles in all journals with MSC (2010): 53C20, 53C22, 53A04, 53A05


Additional Information

Mohammad Ghomi
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
Email: ghomi@math.gatech.edu

DOI: https://doi.org/10.1090/S0002-9939-2010-10507-5
Keywords: Four vertex theorems, flat structures with conical singularities, surfaces of constant curvature with boundary, space forms, geodesic vertices.
Received by editor(s): September 30, 2009
Received by editor(s) in revised form: March 19, 2010
Published electronically: July 22, 2010
Additional Notes: The author was supported by NSF Grant DMS-0336455, and CAREER award DMS-0332333.
Communicated by: Jon G. Wolfson
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society