Ricci Solitons in Low Dimensions

Bennett Chow

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English

American Mathematical Society
30 November 2023

Geometry; Topology

Series: Graduate Studies in Mathematics

Ricci flow is an exciting subject of mathematics with diverse applications in geometry, topology, and other fields. It employs a heat-type equation to smooth an initial Riemannian metric on a manifold. The formation of singularities in the manifold's topology and geometry is a desirable outcome. Upon closer examination, these singularities often reveal intriguing structures known as Ricci solitons.

This introductory book focuses on Ricci solitons, shedding light on their role in understanding singularity formation in Ricci flow and formulating surgery-based Ricci flow, which holds potential applications in topology. Notably successful in dimension 3, the book narrows its scope to low dimensions: 2 and 3, where the theory of Ricci solitons is well established. A comprehensive discussion of this theory is provided, while also establishing the groundwork for exploring Ricci solitons in higher dimensions.

A particularly exciting area of study involves the potential applications of Ricci flow in comprehending the topology of 4-dimensional smooth manifolds. Geared towards graduate students who have completed a one-semester course on Riemannian geometry, this book serves as an ideal resource for related courses or seminars centered on Ricci solitons.

By: Bennett Chow
Imprint: American Mathematical Society
Country of Publication: United States
Dimensions: Height: 254mm, Width: 178mm,
Weight: 408g
ISBN: 9781470474287
ISBN 10: 147047428X
Series: Graduate Studies in Mathematics
Pages: 339
Publication Date: 30 November 2023
Audience: Professional and scholarly , Undergraduate
Format: Hardback
Publisher's Status: Active

Ricci flow singularity formation The Ricci soliton equation The $2$-dimensional classification Estimates for shrinking Ricci solitons Classification of $3$-dimensional shrinkers The Bryant soliton Expanding and steady GRS and the flying wing Brendle's theorem on the uniqueness of $3$-dimensional steadies Geometric preliminaries Analytic preliminaries Bibliography Index

Bennett Chow, University of California, San Diego, La Jolla, CA.