A lower bound to the spectral threshold in curved quantum layers
  We derive a lower bound to the spectral threshold of the Dirichlet Laplacian in tubular neighbourhoods of constant radius about complete surfaces. This lower bound is given by the lowest eigenvalue of a one-dimensional operator depending on the radius and principal curvatures of the reference surface. Moreover, we show that it is optimal if the reference surface is non-negatively curved. 
Pedro Freitas and David Krejcirik
math-ph math.DG math.MP math.SP