Ricci flow on surfaces with conic singularities

Abstract

We establish short-time existence of the Ricci flow on surfaces with a finite number of conic points, all with cone angle between 0 and 2π, with cone angles remain fixed or changing in some smooth prescribed way. For the angle-preserving flow we prove long-time existence; if the angles satisfy the Troyanov condition, this flow converges exponentially to the unique constant curvature metric with these cone angles; if this condition fails, the conformal factor blows up at precisely one point. This is a first step towards the one-dimensional version of the Hamilton-Tian conjecture. These geometric results rely on a new refined regularity theorem for solutions of linear parabolic equations on manifolds with conic singularities. This is proved using methods from geometric microlocal analysis which is the main novelty of this article.

Sharp parabolic regularity and geometric flows on singular spaces

“…We prove in [23] that if β does satisfy the Troyanov condition, then g(t) converges exponentially quickly to the uniformizing constant curvature metric. However, if β does not satisfy the Troyanov condition then one cannot expect such convergence since there is no stationary solution to which g(t) could flow.…”

“…, p k } becomes instantaneously complete. Finally, a main result of [23] states that there is a unique family of conic metrics g(t) for which the cone angles remain constant. Slightly more generally, there is a solution with any prescribed smooth function β(t) of cone angle parameters.…”

“…We also refer to the work of Ramos [31]. We advertise the method of [23] however because it addresses the regularity issues directly, and the asymptotics of solutions obtained this way are used in the long-time existence and convergence steps. These asymptotics will also be helpful for studying further problems, for example the evolution of global spectral invariants under the flow.…”

“…We shall not say much about how these results are obtained, referring to [23] for a careful explanation. The main point, however, is that we rely on a detailed geometric description of the Schwartz kernel of the heat operator.…”

“…The main observation is that when interpreted this way, the underlying PDE is another example of a conic problem. The estimates from [23] may then be invoked almost verbatim to prove the theorem. We note that there is an earlier proof of short-time existence for this general network flow, along with some long-time existence results, due to Ilmanen, Neves and Schulze [15].…”