  Let M be a compact 3-manifold whose interior admits a complete hyperbolic
structure. We let Lambda(M) be the supremum of the bottom eigenvalue of the
Laplacian of N, where N varies over all hyperbolic 3-manifolds homeomorphic to
the interior of M. Similarly, we let D(M) be the infimum of the Hausdorff
dimensions of limit sets of Kleinian groups whose quotients are homeomorphic to
the interior of M. We observe that Lambda(M)=D(M)(2-D(M)) if M is not
handlebody or a thickened torus. We characterize exactly when Lambda(M)=1 and
D(M)=1 in terms of the characteristic submanifold of the incompressible core of
M.
