Ricci flow on asymptotically conical surfaces with nontrivial topology

TL;DR

This paper studies Ricci flow on asymptotically conical surfaces, proving long-time existence, preservation of geometry, and convergence to hyperbolic metrics, revealing the flow's asymptotic behavior.

Contribution

It establishes long-time existence and geometric preservation for Ricci flow on asymptotically conical surfaces, and describes the flow's convergence to hyperbolic metrics.

Findings

Flow preserves asymptotically conical geometry.

The metric expands linearly over time.

Rescaled metrics converge to hyperbolic geometry.

Abstract

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 $g_0$.