Characteristic Property of Delaunay surfaces in S³ and H³
by
L. A. Masaltsev
Kharkov State University, Ukraine

Poster

If a minimal surface in Euclidean space R³ has one family of flat curvature lines, then another family is also flat. It is known that this property have only the Enneper minimal surface, the cathenoid and the family of associated Bonnet surfaces [1, p.544].

Studing the same problem in the hyperbolic space H³ H.C. Wente has shown that for minimal surface x(u, v) in H³ ⊂ R^{3, 1} with fundamental forms ds²=e^{2 w}(du²+dv²), II=-du²+dv² the problem reduces to the solution of the system of two differential equations

Dw = e^2w +e^-2w, m [2, th.5.5], that the only minimal surfaces with flat lines of one family of curvature lines are the surfaces of revolution (the Delaunay surfaces in H³).

A similar question for surfaces of constant mean curvature H in the sphere S³ and in the hyperbolic space H³ which have no umbilical points may be posed. In this case the fundamental forms of studied surface can be presented so: ds²=e^{2 w}(du²+dv²), II=(e^{2 w}H+1)du² +(e^{2 w}H-1)dv². The problem reduces to the solution of the system of two differential equations (first of whose is the Gauss equation and the second gives a condition of flatness of one family of curvature lines);

dw+(H²+c)e^{2 w}-e^{-2 w}=0,

w_uv(He^{2 w} +1)- w_u w_v(He^{2 w} -1)=0

where c=-1 for the case of hyperbolic space H³ and c=1 for the case of sphere S³. Using theorem 2.2 from [2] it is possible to generalize Wente's result in the following way.

Theorem The only cmc surfaces without umbilical points in the hyperbolic space H³ and in the spere S³ which have one family of curvature lines , liyng at the totally geodesic planes, are the surfaces of revolution ( the surfaces of Delaunay).

So the property of flatness of curvature lines of cmc surface without umbilical points characterize the Delaunay surfaces in H³ and S³.

In the case of cmc surfaces in Euclidean space R³ the problem was solved by M.Voretzsch in 1883 (see exposition in [3, ch.10]) and there is a large variety of surfaces with required properties differed from the Delaunay surfaces.

References

1. Bianchi L. Lezioni di geometria differenziale. Vol 1, Bologna, edit. Nic. Zanichelli, 1927.

2. Wente H.C. Constant mean curvature immersions of Enneper type. Memoires Amer. Math. Soc., 1992, No.478.

3. Darboux G. Lecons sur la theorie generale des surfaces. 4 Partie, Paris, Gauthier-Villars, 1925.

L.Masaltsev, Department of Math. and Mechanics, Kharkov state univ., 4 Svobody Sqr., Kharkov, Ukraine; den@itl.net.ua

Date received: April 23, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadq-32.