Wellposedness of nonlinear flows on manifolds of bounded geometry

Abstract

We present simple conditions which ensure that a strongly elliptic operator $L$ generates an analytic semigroup on Hölder 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 $(λI - L)^{-1}$ as $λ$ diverges from the Hölder spectrum of $L$. As one step, we prove existence of this resolvent if $λ$ 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 Hölder spaces of the flow associated to the ambient obstruction tensor on complete manifolds of bounded geometry.