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 $(\lambda I-L{)}^{-1}$ as $\lambda$ diverges from the Hölder 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 Hölder spaces of the flow associated to the ambient obstruction tensor on complete manifolds of bounded geometry.

36 pages