On surfaces of finite type in Euclidean 3-space
type: Item
instance of: scholarly article
title: On surfaces of finite type in Euclidean 3-space
zbMATH Open document ID: 0705.53027
author: Johan Pas
author: Leopold Verstraelen
author: Franki Dillen
published in: Kodai Mathematical Journal
publication date: 1990-01-01T00:00:00Z
review text: Let \(M^ n\) be an n-dimensional submanifold in \(E^ m\). Let x and \(\Delta\) be respectively the position vector field and the Laplace operator of the induced metric on \(M^ n\). \(M^ n\) is said to be of finite type if the spectral decomposition of x on \(M^ n\) is finite, i.e. if (1) \(x=x_ 0+\sum^{q}_{t=p}x_ t\) where \(x_ 0\in {\mathbb{R}}^ m\) is a fixed vector and \(\Delta x_ t=\lambda_ tx_ t\). When there are exactly k non-constant eigenvectors \(x_ t\) which all belong to different eigenvalues \(\lambda_ t\), then \(M^ n\) is said to be of k-type in \(E^ m\). Many results were obtained by \textit{B. Y. Chen} [Bull. Inst. Math., Acad. Sin. 11, 309-328 (1983; Zbl 0498.53039); Bull. Soc. Math. Belg., Ser. B 39, 243-254 (1987; Zbl 0628.53011)]. In the present paper surfaces in \(E^ 3\) which satisfy \((*)\quad \Delta x=Ax+B\) where \(A\in {\mathbb{R}}^{3\times 3}\) and \(B\in {\mathbb{R}}^ 3\) are studied and the following result is obtained; a surface \(M^ 2\) in \(E^ 3\) satisfies (*) if and only if it is an open part of a minimal surface, a sphere or a circular cylinder. This supports a conjecture of B. Y. Chen which states that ordinary spheres are the only compact finite type surfaces in \(E^ 3.\) \{Reviewer's remark: in the sum of (1) t need not be consecutive.\}
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 4157423
zbMATH Keywords: finite type
zbMATH Keywords: spectral decomposition
zbMATH Keywords: minimal surface
zbMATH Keywords: sphere
zbMATH Keywords: circular cylinder
reviewed by: Yosio Mutō
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138039155
OpenAlex ID: W1994024407
cites work: Q3660490
cites work: Q3322795
cites work: Q3764820
cites work: On a certain class of finite type surfaces of revolution
cites work: An extension of Takahashi's theorem
cites work: Minimal immersions of Riemannian manifolds
DOI: 10.2996/KMJ/1138039155
Recommended article: Q3764820
Recommended article: Q4867737
Recommended article: On some \(L_1\)-finite type Euclidean surfaces
Recommended article: Q3480848
Recommended article: Q3806287
Recommended article: Q3357833
Recommended article: A classification of ruled surfaces of finite type in \(S^ 3\)
Recommended article: Surfaces in \(\mathbb{S}^3\) of \(L_1\)-2-type
Recommended article: Q3650020
Recommended article: Q3396108
description: scientific article; zbMATH DE number 4157423