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
DOI: https://doi.org/10.1090/proc/15200
Published electronically: September 4, 2020
MathSciNet review: 4163841
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?)

References
  • A. Cayley, On a special surface of minimum area, Quart. J. P. Appl. Math. 14 (1877), 190–196.
  • C. Delaunay, Sur la surface de révolution dont la courbure moyenne est constante, J. Math. Pures et Appl. 6 (1841), no. 1, 309–320.
  • 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, DOI https://doi.org/10.1007/BF02849386
  • 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, DOI https://doi.org/10.1007/BF02848440
  • Huili Liu, Translation surfaces with constant mean curvature in $3$-dimensional spaces, J. Geom. 64 (1999), no. 1-2, 141–149. MR 1675966, DOI https://doi.org/10.1007/BF01229219
  • 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
  • H. F. Scherk, Bemerkungen über die kleinste Fläche innerhalb gegebener Grenzen, J. Reine Angew. Math. 13 (1835), 185–208 (German). MR 1578041, DOI https://doi.org/10.1515/crll.1835.13.185
  • H. A. Schwarz, Gesammelte mathematische Abhandlungen, 2 vols. Springer, Berlin, 1890. MR0392470 (52 #13287)
  • 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
Email: thasanis@uoi.gr

Rafael López
Affiliation: Departamento de Geometría y Topología, Universidad de Granada, 18071 Granada, Spain
ORCID: 0000-0003-3108-7009
Email: rcamino@ugr.es

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