Heat equation and the Yamabe flow on manifolds with fibered boundary metric

Abstract

This work is dedicated to the study of the Yamabe flow on a class of non-compact complete Riemannian manifolds with fibered boundary and infinite volume, called Phi-manifolds. Some examples of this type of manifold include gravitational instantons, products of an asymptotically conical manifold with a closed manifold, and non-abelian magnetic monopoles. Through assumptions on the regularity of the initial scalar curvature, we prove both the existence and uniqueness of the flow for short time. Moreover, assuming the initial scalar curvature to be negative, bounded, and bounded away from zero, we show that the curvature-normalized flows exist for all time and, further, that they converge to some Riemannian metric with constant scalar curvature.