Asymptotically Flat Ricci Flows

Abstract

We study Ricci flows on $R^n$, $n\ge 3$, that evolve from asymptotically flat initial data. Under mild conditions on the initial data, we show that the flow exists and remains asymptotically flat for an interval of time. The mass is constant in time along the flow. We then specialize to the case of rotationally symmetric, asymptotically flat initial data containing no embedded minimal hyperspheres. We show that in this case the flow is immortal, remains asymptotically flat, never develops a minimal hypersphere, and converges to flat Euclidean space as the time diverges to infinity. We discuss the behaviour of quasi-local mass under the flow, and relate this to a conjecture in string theory.