Well-posedness of nonlinear flows on manifolds of bounded geometry

Abstract

We present straightforward conditions which ensure that a strongly elliptic linear 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. A key step is that we prove existence of this resolvent if \(\lambda \) is sufficiently large using a geometric microlocal version of the semiclassical pseudodifferential calculus. The properties of L and \(e^{-tL}\) we obtain can then be used to prove well-posedness of a wide class of nonlinear flows. We illustrate this by proving well-posedness on Hölder spaces of the flow associated with the ambient obstruction tensor on complete manifolds of bounded geometry. This new result for a higher-order flow on a noncompact manifold exhibits the broader applicability of our technique.