Some classification theorems for submanifolds in Minkowski space-time
The study of submanifolds whose mean curvature vector is an eigenvector of \(\Delta\) was initiated by the author in [Algebra, analysis, and geometry, Proc. Symp. in Honor of Chen-Jung Hsu and Kung-Sing Shih, Taipei/Taiwan 1988, 1-18 (1989; Zbl 0735.53039)] in which the author proved that a submanifold \(M\) of a Euclidean space satisfies the condition \(\Delta H = \lambda H\) for some constant \(\lambda\) if and only if either \(M\) is biharmonic (\(\lambda = 0\)), or \(M\) is of 1-type or of null 2-type. Moreover, he had completely classified surfaces in \(E^ 3\) satisfying \(\Delta H = \lambda H\). In this paper, the author completely classifies submanifolds in the hyperbolic space \(H^{m-1}(-c)\), imbedded standardly in the Minkowski space-time \(E^ m_ 1\), whose mean curvature vector in \(E^ m_ 1\) satisfies the condition \(\Delta H = \lambda H\) for some constant \(\lambda\). He proves that such submanifolds are exactly minimal submanifolds of \(H^{m-1}(-c)\) or minimal submanifolds of a totally umbilical hypersurface of \(H^{m-1}(-c)\). In particular, if \(\lambda = 0\), he obtains the following results: Theorem A. If \(M\) is a non-minimal submanifold of \(H^{m-1}(-1)\), then the following statements are equivalent: (a) \(M\) is a biharmonic submanifold of \(E^ m_ 1\); (b) The mean curvature vector of \(M\) in \(E^ m_ 1\) is a light-like constant vector; (c) \(M\) is a pseudo- umbilical submanifold with unit parallel mean curvature vector in \(H^{m-1}(-1)\); (d) \(M\) is contained in a flat totally umbilical hypersurface of \(H^{m-1}(-1)\) as a minimal submanifold. Theorem B. Let \(M\) be an \(n\)-dimensional, non-minimal, biharmonic submanifold of \(H^{m-1}(-1)\), then (a) the Ricci tensor of \(M\) is negative semi-definite; (b) \(M\) is Ricci flat if and only if \(M\) lies in an \((n+1)\)-dimensional totally geodesic submanifold of \(H^{m-1}(-1)\); (c) \(M\) is Ricci flat if and only if \(M\) is flat; (d) the mean curvature vector of \(M\) in \(E^ m_ 1\) is a constant vector, say c; and \(M\) is contained in the de Sitter space-time \(S^{m-1}_ 1(c,1)\) with \(c\) as its center. Moreover, the positive vector \(x\) of \(M\) in \(E^ m_ 1\) is exactly the mean curvature vector of \(M\) in the de Sitter space-time; (e) \(M\) is contained in the hyperplane \(L\) of \(E^ m_ 1\) defined by \(\langle x,c\rangle = -1\), where \(c\) is given by (d). Similar results in de Sitter space-times are also obtained.
scientific article; zbMATH DE number 743519