The present note reports an optimal bound for a version of the spectral fourth power moment of Hecke $L$-functions associated with Maass forms over the full modular group, in which the spectral parameter runs over short intervals. Consequentially, a new hybrid subconvexity bound is attained for individual values of those $L$-functions on the critical line.

A Laplacian comparison theorem is given. As applications we show a volume comparison theorem and a criterion for the hyperbolicity of Riemannian manifolds.

Let $X$ be a compact 3-manifold and $A$ a boundary component of $X$ to which $X$ retracts. We propose the problem of classifying such $(X,A)$ up to PL homeomorphisms in a suitable class of manifolds. We give a complete solution in a special case. We show in this case there are exactly twenty-four basic pairs $(X,A)$ and all other cases are obtained from these by some obvious modifications. If $A$ is assumed to be different from a 2-sphere, there are six basic pairs.