Extrinsic Geometric Flows

Author : Bennett Chow,Christine Guenther,Mat Langford
Publisher : American Mathematical Soc.
Page : 790 pages
File Size : 40,8 Mb
Release : 2020-05-14
Category : Education
ISBN : 9781470455965

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Extrinsic Geometric Flows by Bennett Chow,Christine Guenther,Mat Langford Pdf

Extrinsic geometric flows are characterized by a submanifold evolving in an ambient space with velocity determined by its extrinsic curvature. The goal of this book is to give an extensive introduction to a few of the most prominent extrinsic flows, namely, the curve shortening flow, the mean curvature flow, the Gauß curvature flow, the inverse-mean curvature flow, and fully nonlinear flows of mean curvature and inverse-mean curvature type. The authors highlight techniques and behaviors that frequently arise in the study of these (and other) flows. To illustrate the broad applicability of the techniques developed, they also consider general classes of fully nonlinear curvature flows. The book is written at the level of a graduate student who has had a basic course in differential geometry and has some familiarity with partial differential equations. It is intended also to be useful as a reference for specialists. In general, the authors provide detailed proofs, although for some more specialized results they may only present the main ideas; in such cases, they provide references for complete proofs. A brief survey of additional topics, with extensive references, can be found in the notes and commentary at the end of each chapter.

Author : Ben Andrews,Bennett Chow,Christine Guenther,Mat Langford
Publisher : American Mathematical Society
Page : 790 pages
File Size : 43,6 Mb
Release : 2022-03-02
Category : Mathematics
ISBN : 9781470464578

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Extrinsic Geometric Flows by Ben Andrews,Bennett Chow,Christine Guenther,Mat Langford Pdf

Extrinsic geometric flows are characterized by a submanifold evolving in an ambient space with velocity determined by its extrinsic curvature. The goal of this book is to give an extensive introduction to a few of the most prominent extrinsic flows, namely, the curve shortening flow, the mean curvature flow, the Gauß curvature flow, the inverse-mean curvature flow, and fully nonlinear flows of mean curvature and inverse-mean curvature type. The authors highlight techniques and behaviors that frequently arise in the study of these (and other) flows. To illustrate the broad applicability of the techniques developed, they also consider general classes of fully nonlinear curvature flows. The book is written at the level of a graduate student who has had a basic course in differential geometry and has some familiarity with partial differential equations. It is intended also to be useful as a reference for specialists. In general, the authors provide detailed proofs, although for some more specialized results they may only present the main ideas; in such cases, they provide references for complete proofs. A brief survey of additional topics, with extensive references, can be found in the notes and commentary at the end of each chapter.

Author : Vladimir Rovenski,Paweł Walczak
Publisher : Springer Nature
Page : 319 pages
File Size : 42,6 Mb
Release : 2021-05-22
Category : Mathematics
ISBN : 9783030700676

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Extrinsic Geometry of Foliations by Vladimir Rovenski,Paweł Walczak Pdf

This book is devoted to geometric problems of foliation theory, in particular those related to extrinsic geometry, modern branch of Riemannian Geometry. The concept of mixed curvature is central to the discussion, and a version of the deep problem of the Ricci curvature for the case of mixed curvature of foliations is examined. The book is divided into five chapters that deal with integral and variation formulas and curvature and dynamics of foliations. Different approaches and methods (local and global, regular and singular) in solving the problems are described using integral and variation formulas, extrinsic geometric flows, generalizations of the Ricci and scalar curvatures, pseudo-Riemannian and metric-affine geometries, and 'computable' Finsler metrics. The book presents the state of the art in geometric and analytical theory of foliations as a continuation of the authors' life-long work in extrinsic geometry. It is designed for newcomers to the field as well as experienced geometers working in Riemannian geometry, foliation theory, differential topology, and a wide range of researchers in differential equations and their applications. It may also be a useful supplement to postgraduate level work and can inspire new interesting topics to explore.

Author : Vicente Cortés,Klaus Kröncke,Jan Louis
Publisher : Springer
Page : 121 pages
File Size : 43,5 Mb
Release : 2018-12-05
Category : Mathematics
ISBN : 9783030011260

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Geometric Flows and the Geometry of Space-time by Vicente Cortés,Klaus Kröncke,Jan Louis Pdf

This book consists of two lecture notes on geometric flow equations (O. Schnürer) and Lorentzian geometry - holonomy, spinors and Cauchy Problems (H. Baum and T. Leistner) written by leading experts in these fields. It grew out of the summer school “Geometric flows and the geometry of space-time” held in Hamburg (2016) and provides an excellent introduction for students of mathematics and theoretical physics to important themes of current research in global analysis, differential geometry and mathematical physics

Author : Vladimir Rovenski,Paweł Walczak
Publisher : Springer Science & Business Media
Page : 129 pages
File Size : 54,6 Mb
Release : 2011-07-26
Category : Mathematics
ISBN : 9781441999085

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Topics in Extrinsic Geometry of Codimension-One Foliations by Vladimir Rovenski,Paweł Walczak Pdf

Extrinsic geometry describes properties of foliations on Riemannian manifolds which can be expressed in terms of the second fundamental form of the leaves. The authors of Topics in Extrinsic Geometry of Codimension-One Foliations achieve a technical tour de force, which will lead to important geometric results. The Integral Formulae, introduced in chapter 1, is a useful for problems such as: prescribing higher mean curvatures of foliations, minimizing volume and energy defined for vector or plane fields on manifolds, and existence of foliations whose leaves enjoy given geometric properties. The Integral Formulae steams from a Reeb formula, for foliations on space forms which generalize the classical ones. For a special auxiliary functions the formulae involve the Newton transformations of the Weingarten operator. The central topic of this book is Extrinsic Geometric Flow (EGF) on foliated manifolds, which may be a tool for prescribing extrinsic geometric properties of foliations. To develop EGF, one needs Variational Formulae, revealed in chapter 2, which expresses a change in different extrinsic geometric quantities of a fixed foliation under leaf-wise variation of the Riemannian Structure of the ambient manifold. Chapter 3 defines a general notion of EGF and studies the evolution of Riemannian metrics along the trajectories of this flow(e.g., describes the short-time existence and uniqueness theory and estimate the maximal existence time).Some special solutions (called Extrinsic Geometric Solutions) of EGF are presented and are of great interest, since they provide Riemannian Structures with very particular geometry of the leaves. This work is aimed at those who have an interest in the differential geometry of submanifolds and foliations of Riemannian manifolds.

Author : Huai-Dong Cao,Shing-Tung Yau
Publisher : Unknown
Page : 368 pages
File Size : 51,8 Mb
Release : 2008
Category : Geometry, Differential
ISBN : STANFORD:36105130567345

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Geometric Flows by Huai-Dong Cao,Shing-Tung Yau Pdf

Author : Manuel Ritoré,Carlo Sinestrari
Publisher : Springer Science & Business Media
Page : 114 pages
File Size : 42,8 Mb
Release : 2010-01-01
Category : Mathematics
ISBN : 9783034602136

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Mean Curvature Flow and Isoperimetric Inequalities by Manuel Ritoré,Carlo Sinestrari Pdf

Geometric flows have many applications in physics and geometry. The mean curvature flow occurs in the description of the interface evolution in certain physical models. This is related to the property that such a flow is the gradient flow of the area functional and therefore appears naturally in problems where a surface energy is minimized. The mean curvature flow also has many geometric applications, in analogy with the Ricci flow of metrics on abstract riemannian manifolds. One can use this flow as a tool to obtain classification results for surfaces satisfying certain curvature conditions, as well as to construct minimal surfaces. Geometric flows, obtained from solutions of geometric parabolic equations, can be considered as an alternative tool to prove isoperimetric inequalities. On the other hand, isoperimetric inequalities can help in treating several aspects of convergence of these flows. Isoperimetric inequalities have many applications in other fields of geometry, like hyperbolic manifolds.

Author : Xi-Ping Zhu
Publisher : American Mathematical Soc.
Page : 162 pages
File Size : 52,6 Mb
Release : 2002
Category : Flows (Differentiable dynamical systems).
ISBN : 9780821833117

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Lectures on Mean Curvature Flows by Xi-Ping Zhu Pdf

``Mean curvature flow'' is a term that is used to describe the evolution of a hypersurface whose normal velocity is given by the mean curvature. In the simplest case of a convex closed curve on the plane, the properties of the mean curvature flow are described by Gage-Hamilton's theorem. This theorem states that under the mean curvature flow, the curve collapses to a point, and if the flow is diluted so that the enclosed area equals $\pi$, the curve tends to the unit circle. In thisbook, the author gives a comprehensive account of fundamental results on singularities and the asymptotic behavior of mean curvature flows in higher dimensions. Among other topics, he considers in detail Huisken's theorem (a generalization of Gage-Hamilton's theorem to higher dimension), evolutionof non-convex curves and hypersurfaces, and the classification of singularities of the mean curvature flow. Because of the importance of the mean curvature flow and its numerous applications in differential geometry and partial differential equations, as well as in engineering, chemistry, and biology, this book can be useful to graduate students and researchers working in these areas. The book would also make a nice supplementary text for an advanced course in differential geometry.Prerequisites include basic differential geometry, partial differential equations, and related applications.

Author : Huai-Dong Cao,Shing-Tung Yau
Publisher : Unknown
Page : 347 pages
File Size : 41,8 Mb
Release : 2008
Category : Electronic
ISBN : 1571461825

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Geometric Flows by Huai-Dong Cao,Shing-Tung Yau Pdf

Author : Paul Baird,Ahmad El Soufi,Ali Fardoun,Rachid Regbaoui
Publisher : Birkhäuser
Page : 158 pages
File Size : 51,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034879682

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Variational Problems in Riemannian Geometry by Paul Baird,Ahmad El Soufi,Ali Fardoun,Rachid Regbaoui Pdf

This book collects invited contributions by specialists in the domain of elliptic partial differential equations and geometric flows. There are introductory survey articles as well as papers presenting the latest research results. Among the topics covered are blow-up theory for second order elliptic equations; bubbling phenomena in the harmonic map heat flow; applications of scans and fractional power integrands; heat flow for the p-energy functional; Ricci flow and evolution by curvature of networks of curves in the plane.

Author : Vladimir Rovenski,Paweł Walczak
Publisher : Springer
Page : 243 pages
File Size : 40,7 Mb
Release : 2014-05-05
Category : Mathematics
ISBN : 9783319046754

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Geometry and its Applications by Vladimir Rovenski,Paweł Walczak Pdf

This volume has been divided into two parts: Geometry and Applications. The geometry portion of the book relates primarily to geometric flows, laminations, integral formulae, geometry of vector fields on Lie groups and osculation; the articles in the applications portion concern some particular problems of the theory of dynamical systems, including mathematical problems of liquid flows and a study of cycles for non-dynamical systems. This Work is based on the second international workshop entitled "Geometry and Symbolic Computations," held on May 15-18, 2013 at the University of Haifa and is dedicated to modeling (using symbolic calculations) in differential geometry and its applications in fields such as computer science, tomography and mechanics. It is intended to create a forum for students and researchers in pure and applied geometry to promote discussion of modern state-of-the-art in geometric modeling using symbolic programs such as MapleTM and Mathematica® , as well as presentation of new results.

Author : David Xianfeng Gu,Emil Saucan
Publisher : CRC Press
Page : 589 pages
File Size : 50,5 Mb
Release : 2023-01-31
Category : Computers
ISBN : 9781000804454

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Classical and Discrete Differential Geometry by David Xianfeng Gu,Emil Saucan Pdf

This book introduces differential geometry and cutting-edge findings from the discipline by incorporating both classical approaches and modern discrete differential geometry across all facets and applications, including graphics and imaging, physics and networks. With curvature as the centerpiece, the authors present the development of differential geometry, from curves to surfaces, thence to higher dimensional manifolds; and from smooth structures to metric spaces, weighted manifolds and complexes, and to images, meshes and networks. The first part of the book is a differential geometric study of curves and surfaces in the Euclidean space, enhanced while the second part deals with higher dimensional manifolds centering on curvature by exploring the various ways of extending it to higher dimensional objects and more general structures and how to return to lower dimensional constructs. The third part focuses on computational algorithms in algebraic topology and conformal geometry, applicable for surface parameterization, shape registration and structured mesh generation. The volume will be a useful reference for students of mathematics and computer science, as well as researchers and engineering professionals who are interested in graphics and imaging, complex networks, differential geometry and curvature.

Author : William M. Goldman
Publisher : American Mathematical Society
Page : 494 pages
File Size : 46,5 Mb
Release : 2022-12-29
Category : Mathematics
ISBN : 9781470471033

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Geometric Structures on Manifolds by William M. Goldman Pdf

The theory of geometric structures on manifolds which are locally modeled on a homogeneous space of a Lie group traces back to Charles Ehresmann in the 1930s, although many examples had been studied previously. Such locally homogeneous geometric structures are special cases of Cartan connections where the associated curvature vanishes. This theory received a big boost in the 1970s when W. Thurston put his geometrization program for 3-manifolds in this context. The subject of this book is more ambitious in scope. Unlike Thurston's eight 3-dimensional geometries, it covers structures which are not metric structures, such as affine and projective structures. This book describes the known examples in dimensions one, two and three. Each geometry has its own special features, which provide special tools in its study. Emphasis is given to the interrelationships between different geometries and how one kind of geometric structure induces structures modeled on a different geometry. Up to now, much of the literature has been somewhat inaccessible and the book collects many of the pieces into one unified work. This book focuses on several successful classification problems. Namely, fix a geometry in the sense of Klein and a topological manifold. Then the different ways of locally putting the geometry on the manifold lead to a “moduli space”. Often the moduli space carries a rich geometry of its own reflecting the model geometry. The book is self-contained and accessible to students who have taken first-year graduate courses in topology, smooth manifolds, differential geometry and Lie groups.

Author : Michael Levitin,Dan Mangoubi,Iosif Polterovich
Publisher : American Mathematical Society
Page : 346 pages
File Size : 47,9 Mb
Release : 2023-11-30
Category : Mathematics
ISBN : 9781470475253

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Topics in Spectral Geometry by Michael Levitin,Dan Mangoubi,Iosif Polterovich Pdf

It is remarkable that various distinct physical phenomena, such as wave propagation, heat diffusion, electron movement in quantum mechanics, oscillations of fluid in a container, can be described using the same differential operator, the Laplacian. Spectral data (i.e., eigenvalues and eigenfunctions) of the Laplacian depend in a subtle way on the geometry of the underlying object, e.g., a Euclidean domain or a Riemannian manifold, on which the operator is defined. This dependence, or, rather, the interplay between the geometry and the spectrum, is the main subject of spectral geometry. Its roots can be traced to Ernst Chladni's experiments with vibrating plates, Lord Rayleigh's theory of sound, and Mark Kac's celebrated question “Can one hear the shape of a drum?” In the second half of the twentieth century spectral geometry emerged as a separate branch of geometric analysis. Nowadays it is a rapidly developing area of mathematics, with close connections to other fields, such as differential geometry, mathematical physics, partial differential equations, number theory, dynamical systems, and numerical analysis. This book can be used for a graduate or an advanced undergraduate course on spectral geometry, starting from the basics but at the same time covering some of the exciting recent developments which can be explained without too many prerequisites.

Author : Michael Artin
Publisher : American Mathematical Society
Page : 104 pages
File Size : 40,9 Mb
Release : 2022-09-21
Category : Mathematics
ISBN : 9781470471118

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Algebraic Geometry by Michael Artin Pdf

This book is an introduction to the geometry of complex algebraic varieties. It is intended for students who have learned algebra, analysis, and topology, as taught in standard undergraduate courses. So it is a suitable text for a beginning graduate course or an advanced undergraduate course. The book begins with a study of plane algebraic curves, then introduces affine and projective varieties, going on to dimension and constructibility. $mathcal{O}$-modules (quasicoherent sheaves) are defined without reference to sheaf theory, and their cohomology is defined axiomatically. The Riemann-Roch Theorem for curves is proved using projection to the projective line. Some of the points that aren't always treated in beginning courses are Hensel's Lemma, Chevalley's Finiteness Theorem, and the Birkhoff-Grothendieck Theorem. The book contains extensive discussions of finite group actions, lines in $mathbb{P}^3$, and double planes, and it ends with applications of the Riemann-Roch Theorem.