Fraternal Twins: The unreasonable resemblance of the Ricci and mean curvature flows

On June 4, 2019, at a conference hosted by the ETH Zürich on geometric analysis and general relativity in honour of Gerhard Huisken's 60th birthday, Richard S. Hamilton presented a lecture enigmatically entitled "Fraternal Twins". In this lecture, he presented an overview of the key historical and mathematical developments in the study of the mean curvature and Ricci flows, emphasizing the striking similarities which consistently occur at a superficial level, but also pointing out the imperfection of these similarities, and some of the analytical differences which lie behind them - much like fraternal twins, the two flows appear very alike at first sight, even though they are by no means identical.

This book provides an introduction to geometric evolution equations through a study of these twin flows. It contains two parts: the first is dedicated to the mean curvature flow and the second to the Ricci flow, though the order does not matter much - each part may be treated entirely independently of the other. On the other hand, once the reader has gained some familiarity with one twin, they will feel at once an uncanny familiarity with the other.

We do not attempt to provide a comprehensive investigation of our twin subjects but rather offer the reader an enticing aperitif, which we hope may whet their appetite for the subject. Each chapter ends with a selection of exercises, and the book would be well-suited to a one or two semester graduate course in geometry, or even an undergraduate seminar course.