Classification of separable surfaces with constant Gaussian curvature
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© Springer-Verlag GmbH Germany, part of Springer Nature 2020
Thomas Hasanis · Rafael López
Classification of separable surfaces with constant Gaussian curvature Received: 13 January 2020 / Accepted: 12 September 2020 Abstract. We classify all surfaces with constant Gaussian curvature K in Euclidean 3-space that can be expressed by an implicit equation of type f (x) + g(y) + h(z) = 0, where f , g and h are real functions of one variable. If K = 0, we prove that the surface is a surface of revolution, a cylindrical surface or a conical surface, obtaining explicit parametrizations of such surfaces. If K = 0, we prove that the surface is a surface of revolution.
1. Introduction and statement of the results The objective of our investigation is the classification of all surfaces with constant Gaussian curvature in Euclidean 3-space that can be expressed by an implicit equation of type f (x) + g(y) + h(z) = 0, where f , g and h are real functions of one variable. Our motivation arises from the classical theory of minimal surfaces. For example, historically the first two minimal surfaces are separable, namely, the catenoid (cosh z)2 = x 2 + y 2 by Euler in 1744, and the helicoid tan(z) = y/x by Meusnier 1776. In 1835, Scherk discovered all minimal surfaces of type z = φ(x)+ψ(y), where φ and ψ are two real functions [8]. Later, Weingarten addressed the classification problem of all minimal surfaces of type f (x) + g(y) + h(z) = 0, realizing that they form a rich and large family of minimal surfaces [10]. For example, this family contains a variety of minimal surfaces given in term of elliptic integrals as well as periodic minimal surfaces such as the Schwarz surfaces of type P and D. In the middle of the past century, Fréchet gave a deep study of these surfaces obtaining examples with explicit parametrizations [2,3]. The reader can see a description of these surfaces in [7, II-5.2]. We introduce the following terminology. Let R3 denote the Euclidean 3dimensional space, that is, the real vector 3-space R3 endowed with the Euclidean metric , = d x 2 + dy 2 + dz 2 , where (x, y, z) stand for the canonical coordinates of R3 . Locally, any surface of R3 is the zero level set F(x, y, z) = 0 of a function F T. Hasanis: Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece. e-mail: [email protected] Rafael López (B): Departamento de Geometría y Topología, Universidad de Granada, 18071 Granada, Spain. e-mail: [email protected] Mathematics Subject Classification: 53A10 · 53C42
https://doi.org/10.1007/s00229-020-01247-6
T. Hasanis, R. López
defined in an open set O ⊂ R3 . Our interest are those surfaces where the function F is a separable function of its variables x, y and z. Definition 1. A (regular) surface S in R3 is said to be separable if can be expressed as (1) S = {(x, y, z) ∈ R3 : f (x) + g(y) + h(z) = 0}. Here f , g and h are smooth functions defined in certain intervals I1 , I2 and I3 of R, respectively. By the regularity of S, f (x)2 + g (y)2 + h (z)2 = 0 for every x ∈ I1 , y ∈ I2 and z ∈
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