Hypersurfaces whose mean curvature function is of finite type (Q1912009)

Let \(M\) be a submanifold in \(\mathbb{R}^n\). Denote by \(g\) the induced metric and by \(\delta\) the Laplacian associated with \(g\). A map \(\varphi: M\to \mathbb{R}\) is called of finite type if it can be written as a sum of a finite number of eigenfunctions of \(\Delta\). The submanifold \(M\) itself is called of finite type [see a.o. \textit{B.-Y. Chen}, Bull. Inst. Math., Acad. Sin. 11, 309-328 (1983; Zbl 0514.53049)] if all its coordinate functions are maps of finite type. In the present paper the author investigates the hypersurfaces in \(\mathbb{R}^{n+1}\) for which the affine mean curvature is of finite type. In particular he investigates which polynomial translation hypersurfaces, i.e. the hypersurfaces which can be written as \(x_{n+1}= P_1 (x_1)+ \cdots+ P_n (x_n)\) with \(P_i\) a polynomial in \(x_i\), which non-conic hyperquadrics and which hypercylinders have an affine mean curvature function of finite type.