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 characteristic property of Delaunay surfaces

Authors: Thomas Hasanis and Rafael López
Journal: Proc. Amer. Math. Soc. 148 (2020), 5291-5298
MSC (2010): Primary 53A10; Secondary 53C42
Published electronically: September 4, 2020
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that Delaunay surfaces, besides the plane and the cate-
noid, are the only surfaces in Euclidean space with nonzero constant mean curvature that can be expressed by an implicit equation of type $ f(x)+g(y)+h(z)=0$, where $ f$, $ g$ and $ h$ are smooth real functions of one variable.

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

  • [1] A. Cayley,
    On a special surface of minimum area,
    Quart. J. P. Appl. Math. 14 (1877), 190-196.
  • [2] C. Delaunay,
    Sur la surface de révolution dont la courbure moyenne est constante,
    J. Math. Pures et Appl. 6 (1841), no. 1, 309-320.
  • [3] Maurice Fréchet, Détermination des surfaces minima du type 𝑎(𝑥)+𝑏(𝑦)=𝑐(𝑧), Rend. Circ. Mat. Palermo (2) 5 (1956), 238–259 (1957) (French, with Esperanto summary). MR 87139,
  • [4] Maurice Fréchet, Détermination des surfaces minima du type 𝑎(𝑥)+𝑏(𝑦)=𝑐(𝑧). II. Quadratures, Rend. Circ. Mat. Palermo (2) 6 (1957), 5–32 (French). MR 95483,
  • [5] Huili Liu, Translation surfaces with constant mean curvature in 3-dimensional spaces, J. Geom. 64 (1999), no. 1-2, 141–149. MR 1675966,
  • [6] Johannes C. C. Nitsche, Lectures on minimal surfaces. Vol. 1, Cambridge University Press, Cambridge, 1989. Introduction, fundamentals, geometry and basic boundary value problems; Translated from the German by Jerry M. Feinberg; With a German foreword. MR 1015936
  • [7] H. F. Scherk, Bemerkungen über die kleinste Fläche innerhalb gegebener Grenzen, J. Reine Angew. Math. 13 (1835), 185–208 (German). MR 1578041,
  • [8] H. A. Schwarz,
    Gesammelte mathematische Abhandlungen,
    2 vols. Springer, Berlin, 1890. MR0392470 (52 #13287)
  • [9] J. Weingarten,
    Ueber die durch eine Gleichung der Form $ \mathfrak{X}+\mathfrak{Y}+\mathfrak{Z}=0$ darstellbaren Minimalflächen,
    Nachr. Königl. Ges. d. Wissensch. Univ. Göttingen (1887), 272-275.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 53A10, 53C42

Retrieve articles in all journals with MSC (2010): 53A10, 53C42

Additional Information

Thomas Hasanis
Affiliation: Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece
MR Author ID: 82090

Rafael López
Affiliation: Departamento de Geometría y Topología, Universidad de Granada, 18071 Granada, Spain

Keywords: Mean curvature, separable surface, Delaunay surface
Received by editor(s): December 9, 2019
Received by editor(s) in revised form: February 22, 2020
Published electronically: September 4, 2020
Additional Notes: The second author was partially supported by the grant no. MTM2017-89677-P, MINECO/AEI/FEDER, UE
Communicated by: Jiaping Wang
Article copyright: © Copyright 2020 American Mathematical Society