  We address an interesting question raised by Dos Santos Ferreira, Kenig and
Salo about regions ${\mathcal R}_g\subset {\mathbb C}$ for which there can be
uniform $L^{\frac{2n}{n+2}}\to L^{\frac{2n}{n-2}}$ resolvent estimates for
$\Delta_g+\zeta$, $\zeta \in {\mathcal R}_g$, where $\Delta_g$ is the
Laplace-Beltrami operator with metric $g$ on a given compact boundaryless
Riemannian manifold of dimension $n\ge3$. This is related to earlier work of
Kenig, Ruiz and the third author for the Euclidean Laplacian, in which case the
region is the entire complex plane minus any disc centered at the origin.
Presently, we show that for the round metric on the sphere, $S^n$, the
resolvent estimates in Ferreira et al, involving a much smaller region, are
essentially optimal. We do this by establishing sharp bounds based on the
distance from $\zeta$ to the spectrum of $\Delta_{S^n}$.
  In the other direction, we also show that the bounds in \cite{Kenig} can be
sharpened logarithmically for manifolds with nonpositive curvature, and by
powers in the case of the torus, ${\mathbb T}^n={\mathbb R}^n/{\mathbb Z}^n$,
with the flat metric. The latter improves earlier bounds of Shen.
  Further improvements for the torus are obtained using recent techniques of
the first author and his work with Guth based on the multilinear estimates of
Bennett, Carbery and Tao. Our approach also allows us to give a natural
necessary condition for favorable resolvent estimates that is based on a
measurement of the density of the spectrum of $\sqrt{-\Delta_g}$, and,
moreover, a necessary and sufficient condition based on natural improved
spectral projection estimates for shrinking intervals.
