A characteristic property of Delaunay surfaces

Hasanis, Thomas; López, Rafael

doi:10.1090/proc/15200

A characteristic property of Delaunay surfaces
HTML articles powered by AMS MathViewer

by Thomas Hasanis and Rafael López PDF

Proc. Amer. Math. Soc. 148 (2020), 5291-5298 Request permission

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

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 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 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 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 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.