  Consider a quantum particle trapped between a curved layer of constant width
built over a complete, non-compact, $\mathcal C^2$ smooth surface embedded in
$\mathbb{R}^3$. We assume that the surface is asymptotically flat in the sense
that the second fundamental form vanishes at infinity, and that the surface is
not totally geodesic. This geometric setting is known as a quantum layer. We
consider the quantum particle to be governed by the Dirichlet Laplacian as
Hamiltonian. Our work concerns the existence of bound states with energy
beneath the essential spectrum, which implies the existence of discrete
spectrum. We first prove that if the Gauss curvature is integrable, and the
surface is weakly $\kappa$-parabolic, then the discrete spectrum is non-empty.
This result implies that if the total Gauss curvature is non-positive, then the
discrete spectrum is non-empty. We next prove that if the Gauss curvature is
non-negative, then the discrete spectrum is non-empty. Finally, we prove that
if the surface is parabolic, then the discrete spectrum is non-empty if the
layer is sufficiently thin.
