Let $M$ be a compact 3-manifold whose interior admits a complete hyperbolic structure. We let $\Lambda(M)$ be the supremum of $\lambda_0(N)$ where $N$ varies over all hyperbolic 3-manifolds homeomorphic to the interior of $N$. 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 $M$.