Geometric Flows

Author : Ben Andrews,Bennett Chow,Christine Guenther,Mat Langford
Publisher : American Mathematical Society
Page : 790 pages
File Size : 50,8 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 : Andrea Braides,Margherita Solci
Publisher : Springer Nature
Page : 134 pages
File Size : 48,5 Mb
Release : 2021-03-23
Category : Mathematics
ISBN : 9783030699178

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Geometric Flows on Planar Lattices by Andrea Braides,Margherita Solci Pdf

This book introduces the reader to important concepts in modern applied analysis, such as homogenization, gradient flows on metric spaces, geometric evolution, Gamma-convergence tools, applications of geometric measure theory, properties of interfacial energies, etc. This is done by tackling a prototypical problem of interfacial evolution in heterogeneous media, where these concepts are introduced and elaborated in a natural and constructive way. At the same time, the analysis introduces open issues of a general and fundamental nature, at the core of important applications. The focus on two-dimensional lattices as a prototype of heterogeneous media allows visual descriptions of concepts and methods through a large amount of illustrations.

Author : Vicente Cortés,Klaus Kröncke,Jan Louis
Publisher : Springer
Page : 121 pages
File Size : 52,9 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 : Bennett Chow,Christine Guenther,Mat Langford
Publisher : American Mathematical Soc.
Page : 790 pages
File Size : 45,6 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 : Lii-Perng Liou
Publisher : Unknown
Page : 138 pages
File Size : 48,9 Mb
Release : 1996
Category : Electronic
ISBN : MINN:31951P00423857Z

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Geometric Flows on Compact Manifolds by Lii-Perng Liou Pdf

Author : Vladimir Rovenski,Paweł Walczak
Publisher : Springer Nature
Page : 319 pages
File Size : 42,8 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 : Manuel Ritoré,Carlo Sinestrari
Publisher : Springer Science & Business Media
Page : 113 pages
File Size : 44,5 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 : Theodora Bourni,Mat Langford
Publisher : Walter de Gruyter GmbH & Co KG
Page : 149 pages
File Size : 47,7 Mb
Release : 2020-12-07
Category : Mathematics
ISBN : 9783110618365

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Mean Curvature Flow by Theodora Bourni,Mat Langford Pdf

With contributions by leading experts in geometric analysis, this volume is documenting the material presented in the John H. Barrett Memorial Lectures held at the University of Tennessee, Knoxville, on May 29 - June 1, 2018. The central topic of the 2018 lectures was mean curvature flow, and the material in this volume covers all recent developments in this vibrant area that combines partial differential equations with differential geometry.

Author : Spiro Karigiannis,Naichung Conan Leung,Jason D. Lotay
Publisher : Springer Nature
Page : 392 pages
File Size : 49,8 Mb
Release : 2020-05-26
Category : Mathematics
ISBN : 9781071605776

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Lectures and Surveys on G2-Manifolds and Related Topics by Spiro Karigiannis,Naichung Conan Leung,Jason D. Lotay Pdf

This book, one of the first on G2 manifolds in decades, collects introductory lectures and survey articles largely based on talks given at a workshop held at the Fields Institute in August 2017, as part of the major thematic program on geometric analysis. It provides an accessible introduction to various aspects of the geometry of G2 manifolds, including the construction of examples, as well as the intimate relations with calibrated geometry, Yang-Mills gauge theory, and geometric flows. It also features the inclusion of a survey on the new topological and analytic invariants of G2 manifolds that have been recently discovered. The first half of the book, consisting of several introductory lectures, is aimed at experienced graduate students or early career researchers in geometry and topology who wish to familiarize themselves with this burgeoning field. The second half, consisting of numerous survey articles, is intended to be useful to both beginners and experts in the field.

Author : Huai-Dong Cao,Shing-Tung Yau
Publisher : Unknown
Page : 368 pages
File Size : 54,8 Mb
Release : 2008
Category : Geometry, Differential
ISBN : STANFORD:36105130567345

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

Author : 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 : Marco Castrillón López,Luis Hernández Encinas,Pedro Martínez Gadea,Ma Eugenia Rosado María
Publisher : Springer
Page : 198 pages
File Size : 52,7 Mb
Release : 2016-06-30
Category : Science
ISBN : 9783319320854

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Geometry, Algebra and Applications: From Mechanics to Cryptography by Marco Castrillón López,Luis Hernández Encinas,Pedro Martínez Gadea,Ma Eugenia Rosado María Pdf

This volume collects contributions written by different experts in honor of Prof. Jaime Muñoz Masqué. It covers a wide variety of research topics, from differential geometry to algebra, but particularly focuses on the geometric formulation of variational calculus; geometric mechanics and field theories; symmetries and conservation laws of differential equations, and pseudo-Riemannian geometry of homogeneous spaces. It also discusses algebraic applications to cryptography and number theory. It offers state-of-the-art contributions in the context of current research trends. The final result is a challenging panoramic view of connecting problems that initially appear distant.

Author : Bennett Chow,Peng Lu,Lei Ni
Publisher : American Mathematical Society, Science Press
Page : 648 pages
File Size : 44,5 Mb
Release : 2023-07-13
Category : Mathematics
ISBN : 9781470473693

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Hamilton’s Ricci Flow by Bennett Chow,Peng Lu,Lei Ni Pdf

Ricci flow is a powerful analytic method for studying the geometry and topology of manifolds. This book is an introduction to Ricci flow for graduate students and mathematicians interested in working in the subject. To this end, the first chapter is a review of the relevant basics of Riemannian geometry. For the benefit of the student, the text includes a number of exercises of varying difficulty. The book also provides brief introductions to some general methods of geometric analysis and other geometric flows. Comparisons are made between the Ricci flow and the linear heat equation, mean curvature flow, and other geometric evolution equations whenever possible. Several topics of Hamilton's program are covered, such as short time existence, Harnack inequalities, Ricci solitons, Perelman's no local collapsing theorem, singularity analysis, and ancient solutions. A major direction in Ricci flow, via Hamilton's and Perelman's works, is the use of Ricci flow as an approach to solving the Poincaré conjecture and Thurston's geometrization conjecture.

Author : Fabrice Baudoin
Publisher : World Scientific
Page : 152 pages
File Size : 54,5 Mb
Release : 2004
Category : Mathematics
ISBN : 9781860944819

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An Introduction to the Geometry of Stochastic Flows by Fabrice Baudoin Pdf

This book aims to provide a self-contained introduction to the local geometry of the stochastic flows associated with stochastic differential equations. It stresses the view that the local geometry of any stochastic flow is determined very precisely and explicitly by a universal formula referred to as the Chen-Strichartz formula. The natural geometry associated with the Chen-Strichartz formula is the sub-Riemannian geometry whose main tools are introduced throughout the text. By using the connection between stochastic flows and partial differential equations, we apply this point of view of the study of hypoelliptic operators written in Hormander's form.

Author : Vladimir Rovenski,Paweł Walczak
Publisher : Springer Science & Business Media
Page : 129 pages
File Size : 42,5 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.