  We prove $L^p\to L^{p'}$ bounds for the resolvent of the Laplace-Beltrami
operator on a compact Riemannian manifold of dimension $n$ in the endpoint case
$p=2(n+1)/(n+3)$. It has the same behavior with respect to the spectral
parameter $z$ as its Euclidean analogue, due to Kenig-Ruiz-Sogge, provided a
parabolic neighborhood of the positive half-line is removed. This is region is
optimal, for instance, in the case of a sphere.
