Ricci flow on asymptotically conical surfaces with nontrivial topology

Details Ricci flow on asymptotically conical surfaces with nontrivial topology Ricci flow on asymptotically conical surfaces with nontrivial topology

2010-03-26

arxiv.org/abs/1003.5237v1

Mathematics

Differential Geometry

Scientific paper

As part of the general investigation of Ricci flow on complete surfaces with finite total curvature, we study this flow for surfaces with asymptotically conical (which includes as a special case asymptotically Euclidean) geometries. After establishing long-time existence, and in particular the fact that the flow preserves the asymptotically conic geometry, we prove that the solution metric $g(t)$ expands at a locally uniform linear rate; moreover, the rescaled family of metrics $t^{-1}g(t)$ exhibits a transition at infinite time inasmuch as it converges locally uniformly to a complete, finite area hyperbolic metric which is the unique uniformizing metric in the conformal class of the initial metric $g_0$.

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