On surfaces of finite type in Euclidean 3-space
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.\}
scientific article; zbMATH DE number 4157423
scientific article