  Let $M$ be a complete, non-compact, connected Riemannian manifold with Ricci
curvature bounded from below by a negative constant. A sufficient condition is
obtained for open and connected sets $D$ in $M$ for which the corresponding
Dirichlet heat semigroup is intrinsically ultracontractive. That condition is
formulated in terms of capacitary width. It is shown that both the reciprocal
of the bottom of the spectrum of the Dirichlet Laplacian acting in $L^2(D)$,
and the supremum of the torsion function for $D$ are comparable with the square
of the capacitary width for $D$ if the latter is sufficiently small. The
technical key ingredients are the volume doubling property, the Poincar\'e
inequality and the Li-Yau Gaussian estimate for the Dirichlet heat kernel for
finite scale.
