Extrinsic Geometric Flows

Last updated November 19, 2022

Extrinsic Geometric Flows is an advanced mathematics textbook that overviews geometric flows, mathematical problems in which a curve or surface moves continuously according to some rule. It focuses on extrinsic flows, in which the rule depends on the embedding of a surface into space, rather than intrinsic flows such as the Ricci flow that depend on the internal geometry of the surface and can be defined without respect to an embedding.

Topics

The book consists of four chapters, roughly divided into four sections:^[1]

Chapters 1 through 4 concern the heat equation and the curve-shortening flow defined from it, in which a curve moves in the Euclidean plane, perpendicularly to itself, at a speed proportional to its curvature.^[1] It includes material on curves that remain self-similar as they flow, such as circles and the grim reaper curve $y=-\log \cos x$ , the Gage–Hamilton–Grayson theorem according to which every simple closed curve converges to a circle until eventually collapsing to a point, without ever self-intersecting, and the classification of ancient solutions of the flow.^[2]^[3]
Chapters 5 through 14 concern the mean curvature flow, a higher dimensional generalization of the curve-shortening flow that uses the mean curvature of a surface to control the speed of its perpendicular motion.^[1] After an introductory chapter on the geometry of hypersurfaces,^[3] It includes results of Ecker and Huisken concerning "locally Lipschitz entire graphs", and Huisken's theorem that uniformly convex surfaces remain smooth and convex, converging to a sphere, before they collapse to a point.^[2] Huisken's monotonicity formula is covered, as are the regularity theorems of Brakke and White according to which the flow is almost-everywhere smooth.^[3] Several chapters in this section concern the singularities that can develop in this flow, as well as the surfaces that remain self-similar as they flow.^[2]
Chapters 15 through 17 concern the Gauss curvature flow, a different way of generalizing the curve-shortening flow to higher dimensions using Gaussian curvature in place of mean curvature. Although Gaussian curvature is intrinsic, unlike mean curvature, the Gauss curvature flow is extrinsic, because it involves the motion of an embedded surface.^[1] Here, variations of the flow involve using a power of the curvature, rather than the curvature itself, to define the speed of the flow, and this raises questions concerning the existence of the flow over finite time intervals, the existence of self-similar solutions, and limiting shapes.^[2] The exponent of the curvature is critical here, with convex surfaces converging to an ellipsoid at exponent ${\tfrac {1}{d+2}}$ (generizing the affine curve-shortening flow) and to a round sphere for larger exponents.^[3]
Chapters 18-20 provide a broader panorama of nonlinear geometric flows.^[1]

The content within each chapter includes both proofs of the results discussed in the chapter, and references to the mathematics literature; additional references are provided in a commentary section at the end of each chapter, which also provides additional intuition and descriptions of open problems,^[1] as well as brief descriptions of additional results in the same area.^[3] As well as illustrating the mathematics under discussion with many figures,^[4] it humanizes the content by providing photographs of many of the mathematicians that it references.^[1]^[2]^[4] The chapters include exercises, making this book suitable as a graduate textbooks.^[1]

Audience and reception

Although intrinsic flows have been the subject of much recent attention in mathematics after their use by Grigori Perelman to solve both the Poincaré conjecture and the geometrization conjecture, extrinsic flows also have a long history of important applications in mathematics, closely related to the solutions of partial differential equations. Their uses include modeling the growth of biological cells, metallic crystal grains, bubbles in foams,^[4] and even "the deformation of rolling stones in a beach".^[3]

The book's proofs are often simplifications of the proofs in the research literature, but nevertheless it still quite technical, aimed at graduate students and researchers in geometric analysis. Readers are expected to be familiar with the basics of differential geometry and partial differential equations.^[1]^[2] There is more material in the book than could be covered in a single course, but it could either form the basis of a multi-course sequence or a topics course that picks out only some of its material.^[4] As well as being a textbook, Extrinsic Geometric Flows can serve as reference material on flows for specialists in the area.^[1]

Related works

This is not the first book on geometric flows. Others include:^[4]

Chou, Kai-Seng; Zhu, Xi-Ping (2001), The Curve Shortening Problem, Chapman & Hall/CRC, doi:10.1201/9781420035704, ISBN 1-58488-213-1, MR 1888641
Zhu, Xi-Ping (2002), Lectures on Mean Curvature Flows, AMS/IP Studies in Advanced Mathematics, vol. 32, International Press, doi:10.1090/amsip/032, ISBN 0-8218-3311-1, MR 1931534
Ecker, Klaus (2004), Regularity Theory for Mean Curvature Flow, Progress in Nonlinear Differential Equations and their Applications, vol. 57, Birkhäuser, doi:10.1007/978-0-8176-8210-1, ISBN 0-8176-3243-3, MR 2024995
Giga, Yoshikazu (2006), Surface Evolution Equations: A Level Set Approach, Monographs in Mathematics, vol. 99, Birkhäuser, ISBN 978-3-7643-2430-8, MR 2238463

Although Extrinsic Geometric Flows is more comprehensive and up-to-date than these works, it omits some of their topics, including anisotropic flows of curves in Chou & Zhu (2001), applications to the theory of relativity in Zhu (2002), and the level-set methods of Giga (2006).^[4]

Related Research Articles

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In the field of differential geometry in mathematics, mean curvature flow is an example of a geometric flow of hypersurfaces in a Riemannian manifold. Intuitively, a family of surfaces evolves under mean curvature flow if the normal component of the velocity of which a point on the surface moves is given by the mean curvature of the surface. For example, a round sphere evolves under mean curvature flow by shrinking inward uniformly. Except in special cases, the mean curvature flow develops singularities.

In the mathematical field of differential geometry, a geometric flow, also called a geometric evolution equation, is a type of partial differential equation for a geometric object such as a Riemannian metric or an embedding. It is not a term with a formal meaning, but is typically understood to refer to parabolic partial differential equations.

In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: extrinsically, relating to their embedding in Euclidean space and intrinsically, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space.

In the mathematical fields of differential geometry and geometric analysis, inverse mean curvature flow (IMCF) is a geometric flow of submanifolds of a Riemannian or pseudo-Riemannian manifold. It has been used to prove a certain case of the Riemannian Penrose inequality, which is of interest in general relativity.

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In differential geometry, the Angenent torus is a smooth embedding of the torus into three-dimensional Euclidean space, with the property that it remains self-similar as it evolves under the mean curvature flow. Its existence shows that, unlike the one-dimensional curve-shortening flow, the two-dimensional mean-curvature flow has embedded surfaces that form more complex singularities as they collapse.

In the mathematical fields of differential geometry and geometric analysis, the Gauss curvature flow is a geometric flow for oriented hypersurfaces of Riemannian manifolds. In the case of curves in a two-dimensional manifold, it is identical with the curve shortening flow. The mean curvature flow is a different geometric flow which also has the curve shortening flow as a special case.

References

1 2 3 4 5 6 7 8 9 10 Ross, John (January 2021), "Review of Extrinsic Geometric Flows", MAA Reviews, Mathematical Association of America
1 2 3 4 5 6 Urbas, John, "Review of Extrinsic Geometric Flows", zbMATH , Zbl 1475.53002
1 2 3 4 5 6 Silva Neto, Gregório Manoel, "Review of Extrinsic Geometric Flows", MathSciNet , MR 4249616
1 2 3 4 5 6 Ni, Lei (2022), "Review of Extrinsic Geometric Flows", Bulletin of the American Mathematical Society, New Series, 59 (1): 145–154, doi: 10.1090/bull/1740 , MR 4347206, Zbl 1484.00045

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[ross-1] 1 2 3 4 5 6 7 8 9 10 Ross, John (January 2021), "Review of Extrinsic Geometric Flows", MAA Reviews, Mathematical Association of America

[urbas-2] 1 2 3 4 5 6 Urbas, John, "Review of Extrinsic Geometric Flows", zbMATH , Zbl 1475.53002

[silva-neto-3] 1 2 3 4 5 6 Silva Neto, Gregório Manoel, "Review of Extrinsic Geometric Flows", MathSciNet , MR 4249616

[ni-4] 1 2 3 4 5 6 Ni, Lei (2022), "Review of Extrinsic Geometric Flows", Bulletin of the American Mathematical Society, New Series, 59 (1): 145–154, doi: 10.1090/bull/1740 , MR 4347206, Zbl 1484.00045

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