Book Review: Ricci flow for shape analysis and surface registration: theories, algorithms and applications

Abstract

One of the goals in the field of modern geometric analysis is to study canonical structures on manifolds and vector bundles-existence, uniqueness, and moduli. One of the pioneers of analytical approaches in this field is Yau, who, with Schoen and other coauthors solved a number of outstanding geometric problems, many of which are related to elliptic PDEs. A personal account of these developments is in the two-volume set [65]. Around the same time, there were remarkable developments in other areas of geometry. A more geometric theory, including the fundamental theories of compactness and collapse, was developed by Cheeger, Gromov, and others. Thurston revolutionized low-dimensional topology by developing a geometric theory aimed at understanding 3-manifolds via the geometrization conjecture and related ideas. Soon after, inspired by the earlier work of Eells and Sampson on harmonic maps, Hamilton invented the Ricci flow, a parabolic PDE, and through a number of very original works nearly single-handedly developed it into a compelling program to approach the Poincaré and geometrization conjectures. Two decades later, Perelman solved the Poincaré and geometrization conjectures by introducing a plethora of deep and powerful ideas and methods into Ricci flow to complete Hamilton's program. At the present time, more than another decade later, geometric analysis is a thriving field with many active subfields, including Kähler geometry, influenced by the work of Tian, Donaldson, Chen, and others, the study of Ricci curvature by Cheeger, Colding, Minicozzi, and others, geometric flows such as the mean curvature flow by Huisken, White, and others, and the work of Brendle to solve long-standing conjectures, to just name a few of the many directions this field has taken.The idea that a discrete set of values determines geometry is the main tenet of discrete differential geometry. These ideas come from a number of disparate directions. Whitney used numerical analysis to connect de Rham theory and simplicial homology [64], and these ideas form the central tenets of discrete exterior calculus [20]. In a separate intellectual stream, Regge introduced a notion of discrete geometry that he considered a way of developing computational and quantum gravity from first principles that were motivated by, but not dependent on, the theory of smooth manifolds (see [34,51]). Thurston introduced the idea of circle packings for approximating Riemann mappings, and this developed into a whole field of circle packing and related topics (see Stephenson's book [58], reviewed in [6]). A discrete form of integrable systems was developed by Bobenko and Suris [5] into a geometric theory, while other groups [1,23] have developed similar or competing theories arising from numerical analysis and mathematical physics. All of these directions may be characterized as structure-preserving discretizations.