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 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.

Recognition of the likeness of the two flows goes back much further, of course. Indeed, the drawing of parallels between the two flows is now customary amongst experts; it is often exclaimed, for instance, that “Ricci flow is the extrinsic analogue of mean curvature flow”, or that “mean curvature flow in n-dimensions behaves like Ricci flow in 2n-dimensions”, or “since P holds for mean curvature flow/Ricci flow, ˜P must be true for Ricci flow/mean curvature flow”. And the comparison is more than superficial: despite the fact that the two flows continue to be treated independently, often with quite different tools, Hamilton’s analogy continues to be vindicated.

The aim of this book is to provide 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 treatment 2 of our twin subjects but rather offer the reader an enticing aperitif, which we hope may whet their appetite for the subject. 3 Each part begins with The fundamentals, introducing the reader to each twin, followed by a technical chapter which lays The groundwork for further analysis. This second chapter could be skipped on first reading, and referred back to as needed in the later chapters; on the other hand, the patient reader will certainly benefit in the long run from any effort put into the groundwork. The third chapter of each part is concerned with curvature Pinching and its consequences, with a focus on the first major milestone in each of our twin subjects—Huisken’s theorem on the contraction of convex hypersurfaces to round points under mean curvature flow and Hamilton’s theorem on the contraction of threemanifolds of positive Ricci curvature to round points under Ricci flow, respectively. We then study each flow in its smallest nontrivial dimension, where the behaviour is particularly nice. The fifth chapter introduces the reader to a selection of tools and results pertaining to Singularities and their analysis for the respective flow (in higher dimensions). We conclude by surveying some of the recent progress Towards a classification of ancient solutions to each flow.

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 “special topics” course. For a one semester course, one could plausibly cover, e.g., Chapters 1-5, or Chapters 7-11, or selected parts of Chapters 1-4 and 7-10.

The project grew out of notes for a minicourse on the Ricci flow which I presented in a series of lectures at the summer school “Geometric Flows and Relativity” hosted by the Centro de Matemática of the Universidad de la República in Montevideo, Uruguay, in March 2024, which were subsequently used in a special topics course on both the mean curvature and Ricci flows aimed at advanced undergraduate and beginning graduate students at The Australian National University. I am grateful to Theodora Bourni and Martín Reiris for the invitation to speak at the CMAT summer school, and to the outstanding cohort of students who attended my lectures, keeping me on my toes each morning; I am equally grateful to my wife, Kirsty, who—heavily pregnant with our second child—encouraged me to go!

Many individuals have contributed to this book through useful discussions, particularly Ben Andrews, Theodora Bourni, Tim Buttsworth, Bennett Chow, Apostolos Damialis, Ramiro Lafuente, Stephen Lynch, Martín Reiris and Jonathan Zhu.

I do not claim priority for any of the mathematical results presented herein, and have endeavoured to provide appropriate bibliographic information throughout. The manuscript was compiled on Overleaf in A Tufte-LTEX and the cover was designed using Adobe Illustrator and Adobe Express. Illustrations were created using GeoGebra and Mathematica. No AI tools were used in any stage of the preparation.

Notes: 1 Some brave souls even speculate that there is a hidden canonical correspondence between the two; but no such correspondence is yet to be observed.

2 This would take up many volumes, and has already been achieved, to a large degree, by others.

3 Incidentally, we heartily recommend a glass of Glenlivet (Founder’s Reserve) to accompany this text, not least because “Glenlivet” may be translated as “Valley of the smooth flow”.  

Mat Langford, Canberra, August 16, 2025