Bennet Chow: “Ricci Solitons in Low Dimensions”. AMS 2023, 339 pp

Bennet Chow's 'Ricci Solitons in Low Dimensions' delves into the intricate world of Ricci solitons, special geometric structures on manifolds that play a crucial role in the singularity formation of Ricci flows. The book focuses on dimensions two and three, explaining how these solitons appear in the evolution of Ricci flows and their classification. It also discusses the heuristics behind the defining equation of the Ricci flow, which was introduced by Hamilton, and highlights important results proven using this flow. Ricci solitons are self-similar solutions of the Ricci flow, evolving along homotheties, and are categorized as shrinking, expanding, or steady solitons. The book provides detailed examples and analyses of these solitons, including the Gaussian shrinker and expander, and the Bryant soliton. It also explores the blowup analysis of singularities in the Ricci flow, distinguishing between Type I and Type II singularities and their blowup limits. The author presents significant results, such as Hamilton and Chow's proof that every compact two-dimensional surface evolves to a surface of constant curvature under the Ricci flow, and Perelman's work on the geometrization conjecture. Additionally, the book covers recent breakthroughs in dimensions, including Bamler and Kleiner's unique Ricci flow through singularities and Bamler's work on singularities in dimension four. The text is accessible to advanced graduate students and researchers, offering a thorough overview of the state-of-the-art research on Ricci solitons, complete with proofs and exercises.