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.

Arxiv version contains Appendices and additional material on local (in time) existence which does not appear in the published version. This version corrects minor errors in equations 5.13 and 5.14. 47 pages

References in corpus (1)

• Rough Solutions of the Einstein Constraint Equations

Cited by in corpus (7)

• Ricci solitons, Ricci flow, and strongly coupled CFT in the Schwarzschild Unruh or Boulware vacua
• On the positive mass theorem for manifolds with corners
• Entanglement entropy and the Ricci flow
• Energy, entropy and the Ricci flow
• On long-time existence for the flow of static metrics with rotational symmetry
• A Note on Conformal Ricci Flow
• The Ricci flow of the `RP3 geon' and noncompact manifolds with essential minimal spheres