Ricci Flow in Two Dimensions

TL;DR

This paper reviews Ricci flow on two-dimensional surfaces, highlighting classical results and recent discoveries on open surfaces, including phenomena observed in complete and incomplete cases.

Contribution

It provides a comprehensive survey of Ricci flow in 2D, emphasizing new phenomena on open surfaces not covered in earlier work.

Findings

Classical results by Hamilton and Chow summarized

Recent work reveals striking new phenomena on open surfaces

Analysis includes both complete and incomplete surfaces

Abstract

Ricci flow on two dimensional surfaces is far simpler than in the higher dimensional cases. This presents an opportunity to obtain much more detailed and comprehensive results. We review the basic facts about this flow, including the original results by Hamilton and Chow concerning Ricci flow on compact surfaces. The rationale for this paper, however, is especially to survey recent work concerning this flow on open surfaces, including various classes of both complete and incomplete surfaces, where a number of striking new phenomena have been observed.