Extrinsic Geometric Flows

This textbook, written by four experts in the field offers an authoritative introduction and overview to the topic of extrinsic geometric flows. It will serve well as a primary text for a graduate student who already has background knowledge of differential geometry and (some) partial differential equations. It will also serve as a useful reference for experts in the field. 

 

Geometric flows are a field of study that sits at the intersection of differential equations and geometry. Such flows are characterized by geometric objects that deform in ways governed by geometric quantities (such as length, area, volume, and curvature). Extrinsic flows are examples of flows in which the geometric object of interest is immersed or embedded in a higher-dimensional ambient space, and then deformed due to extrinsic curvature measurements. 

 

The book is extremely well organized, and is loosely divided into five main sections.  The first section (Ch. 1) consists of a lone chapter covering the classical heat equation on Euclidean space. The heat equation serves as a model for future geometric curvature flows, and the text goes into detail discussing classical results that will be built on in future sections. The next three sections constitute the “meat” of the textbook and cover three major extrinsic flows: the Curve Shortening Flow (Ch. 2-4), the Mean Curvature Flow (Ch. 5-14), and the Gauss Curvature Flow (Ch. 15-17). The text concludes with a section covering additional, fully nonlinear curvature flows (Ch. 18-20). Each of the three major sections presents an overview of the main results for the corresponding flow. As such, each section could serve as a standalone resource. However, the text really shines at drawing comparisons and pointing out similar techniques across the different flows. The authors do a fantastic job of identifying tricks and techniques, explicitly showing how these techniques are modified and used in different settings and for various flows. The groundwork here is set in the first chapter, where tools used to explore the heat equation (monotonicity formulae; maximum principles; instantaneous smoothing; and long-term behavior, to name a few) are introduced, and then appear again and again in modified forms later in the text. The clear connections drawn across sections make this text, when taken as a whole, greater than the sum of its parts.

 

Each section is thorough, offering many full proofs and extensive references to papers both classic and recent. Charmingly, the text includes images of many of the mathematicians cited. Chapters are bolstered by their concluding subsections, titled “Notes and Commentary,” that offer additional context, informal commentary, and occasionally conjectures and open problems in the field. These features -- the citations, headshots, additional notes, and discussion of open problems -- serve to orient the reader to the current state of the field and to (past and present) researchers. Exercises are also included in each chapter. 

 

All told, Extrinsic Geometric Flows is a well-written and comprehensive textbook that will serve as a valuable resource for interested parties, from graduate students up to content experts.

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John Ross is an assistant professor of mathematics at Southwestern University.