Wellposedness of nonlinear flows on manifolds of bounded geometry

TL;DR

This paper establishes conditions under which nonlinear flows on manifolds of bounded geometry are well-posed by analyzing elliptic operators and their resolvents, using microlocal analysis and sectorial operator theory.

Contribution

It introduces simple criteria for elliptic operators to generate analytic semigroups on manifolds of bounded geometry, enabling well-posedness results for nonlinear flows.

Findings

Proved sectoriality of elliptic operators on manifolds of bounded geometry.

Established existence of resolvent operators for large spectral parameters.

Applied results to prove well-posedness of flows related to the ambient obstruction tensor.

Abstract

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.