  We present simple conditions which ensure that a strongly elliptic operator
$L$ generates an analytic semigroup on H\"older spaces on an arbitrary complete
manifold of bounded geometry. This is done by establishing the equivalent
property that $L$ is "sectorial", a condition that specifies the decay of the
resolvent $(\lambda I - L)^{-1}$ as $\lambda$ diverges from the H\"older
spectrum of $L$. As one step, we prove existence of this resolvent if $\lambda$
is sufficiently large, and on this general class of manifolds, use a geometric
microlocal version of the semiclassical pseudodifferential calculus. The
properties of $L$ and $e^{-tL}$ we obtain can then be used to prove
wellposedness of a wide class of nonlinear flows. We illustrate this by proving
wellposedness on H\"older spaces of the flow associated to the ambient
obstruction tensor on complete manifolds of bounded geometry.
