Ricci flow on surfaces with conic singularities

TL;DR

This paper proves short-time existence of Ricci flow on surfaces with conic singularities and shows convergence to constant curvature metrics under certain conditions, extending the understanding of geometric flows on singular surfaces.

Contribution

It establishes the existence and convergence of Ricci flow on surfaces with conic points, including cases with fixed or smoothly changing cone angles, and relates stability conditions to flow behavior.

Findings

Short-time existence of Ricci flow on conic surfaces.

Long-time convergence to constant curvature metrics under angle-preserving flow.

Flow convergence to solitons when stability conditions are not met.

Abstract

We establish the short-time existence of the Ricci flow on surfaces with a finite number of conic points, all with cone angle between 0 and $2\pi$, where the cone angles remain fixed or change in some smooth prescribed way. For the angle-preserving flow we prove long-time existence and convergence. When the Troyanov angle condition is satisfied (equivalently, when the data is logarithmically K-stable), the flow converges to the unique constant curvature metric with the given cone angles; if this condition is not satisfied, the flow converges subsequentially to a soliton. This is the one-dimensional version of the Hamilton--Tian conjecture.