Minimal submanifolds of Euclidean spaces are contained not only in larger classes of submanifolds—for example, in the class of submanifolds of finite type—but also in the class of submanifolds satisfying ΔH = λH, λ ∈ R. The study of sub-manifolds satisfying ΔH = λH was initiated by B.-Y. Chen in 1988, and arose in the context of his theory of submanifolds of finite type. For a survey of recent results on submanifolds of finite type and various related topics, see for example. Let M~n be an n-dimensional connected submanifold of the Euclidean space E~m. Denote by x, H, and Δ (respectively) the position vector field of M~n, the mean curvature vector field of M~n, and the Laplace operator on M~n, with respect to the Riemannian metric g on M~n, induced from the Euclidean metric of the ambient space E~m.
展开▼
