论文信息 - On Closed Weingarten Surfaces

On Closed Weingarten Surfaces

Abstract.We investigate closed surfaces in Euclidean 3-space satisfying certain functional relations κ = F(λ) between the principal curvatures κ, λ. In particular we find analytic closed surfaces of genus zero where F is a quadratic polynomial or F(λ) = cλ2n+1. This generalizes results by H. Hopf on the case where F is linear and the case of ellipsoids of revolution where F(λ) = cλ3.

Wolfgang Kühnel | W. Kühnel | Michael Steller | M. Steller

[1] Nicolaos Kapouleas,et al. Constant mean curvature surfaces in Euclidean three-space , 1987 .

[2] J. Weingarten,et al. Ueber eine Klasse auf einander abwickelbarer Flächen. , 1861 .

[3] Nicolaos Kapouleas. Complete constant mean curvature surfaces in euclidean three-space , 1990 .

[4] Henry C. Wente. Counterexample to a conjecture of H. Hopf , 1986 .

[5] B. van-Brunt,et al. Hyperbolic Weingarten surfaces , 1994, Mathematical Proceedings of the Cambridge Philosophical Society.

[6] W. Kühnel. Differential Geometry: Curves - Surfaces - Manifolds , 2002 .

[7] H. Hopf. Über Flächen mit einer Relation zwischen den Hauptkrümmungen. Meinem Lehrer Erhard Schmidt in Verehrung und Freundschaft zum 75. Geburtstag gewidmet. , 1950 .

[8] K. Voss. Über geschlossene Weingartensche Flächen , 1959 .

[9] Bruce van Brunt,et al. Potential applications of Weingarten surfaces in CAGD, Part I: Weingarten surfaces and surface shape investigation , 1996, Comput. Aided Geom. Des..

[10] S. Chern. Some new characterizations of the Euclidean sphere , 1945 .

[11] W. Kühnel. Zur inneren Krümmung der zweiten Grundform , 1981 .