Existence of Bound States for Layers Built Over Hypersurfaces of Euclidean Space

In this paper, we study the bound states of quantum layers. We prove that for the quantum layer built over a parabolic manifold which is not totally geodesic, if the second fundamantal form decays sufficiently fast, then the bound states exist. In the 2d case, we prove that the quantum layer over a convex surface whose second fundamental form tends to zero at infinity must have bound states.