Compression theorems for surfaces and their applications (Q996134)

Let \(x: M^m \to E^n\), \(m<n\) be an embedding of a Riemannian manifold into the standard Euclidean space. Let \(\vec H\) be the mean curvature vector field. We say that the submanifold \(x(M^m)\) is a submanifold with proper mean curvature vector field if the following equation holds: \[ \Delta \vec H = \lambda \vec H. \] It was proved before that in the case \(n=4,m=3\) the submanifold with proper mean curvature vector field has constant mean curvature. In the paper this result is generalized for the case of isometric embeddings into pseudo-Euclidean space (under the additional condition that the embedding is an embedding with diagonal shape operator).