Smoothening cone points with Ricci flow

TL;DR

This paper demonstrates that Ricci flow can instantaneously smooth out cone points on closed surfaces, extending the method to any cone angle between 0 and 2π, including cusps as a limiting case.

Contribution

It proves the existence of a Ricci flow that smooths cone points on closed surfaces for any cone angle between 0 and 2π, generalizing previous results to include cusps.

Findings

Ricci flow smooths cone points instantaneously.

The method applies to cone angles from 0 to 2π.

Cusps are treated as a limiting case of cone points.

Abstract

We consider Ricci flow on a closed surface with cone points. The main result is: given a (nonsmooth) cone metric g_0 over a closed surface there is a smooth Ricci flow g(t) defined for (0,T], with curvature unbounded above, such that g(t) tends to g_0 as t tends to 0. This result means that Ricci flow provides a way for instantaneously smoothening cone points. We follow an argument of P. Topping modifying his reasoning for cusps of negative curvature; in that sense we can consider cusps as a limiting zero-angle cone, and we generalize to any angle between 0 and 2\pi.