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Brakke's Mean Curvature Flow: An Introduction (SpringerBriefs in Mathematics) 🔍
Yoshihiro Tonegawa Springer Singapore : Imprint: Springer, SpringerBriefs in Mathematics, SpringerBriefs in Mathematics, 1, 2019
英语 [en] · PDF · 1.9MB · 2019 · 📘 非小说类图书 · 🚀/lgli/lgrs/nexusstc/scihub/upload/zlib · Save
描述
This book explains the notion of Brakke’s mean curvature flow and its existence and regularity theories without assuming familiarity with geometric measure theory. The focus of study is a time-parameterized family of __k__-dimensional surfaces in the __n__-dimensional Euclidean space (1 ≤ __k__ < __n__). The family is the mean curvature flow if the velocity of motion of surfaces is given by the mean curvature at each point and time. It is one of the simplest and most important geometric evolution problems with a strong connection to minimal surface theory. In fact, equilibrium of mean curvature flow corresponds precisely to minimal surface. Brakke’s mean curvature flow was first introduced in 1978 as a mathematical model describing the motion of grain boundaries in an annealing pure metal. The grain boundaries move by the mean curvature flow while retaining singularities such as triple junction points. By using a notion of generalized surface called a varifold from geometric measure theory which allows the presence of singularities, Brakke successfully gave it a definition and presented its existence and regularity theories. Recently, the author provided a complete proof of Brakke’s existence and regularity theorems, which form the content of the latter half of the book. The regularity theorem is also a natural generalization of Allard’s regularity theorem, which is a fundamental regularity result for minimal surfaces and for surfaces with bounded mean curvature. By carefully presenting a minimal amount of mathematical tools, often only with intuitive explanation, this book serves as a good starting point for the study of this fascinating object as well as a comprehensive introduction to other important notions from geometric measure theory.
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lgli/N:\!genesis_\0day\springer\10.1007%2F978-981-13-7075-5.pdf
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scihub/10.1007/978-981-13-7075-5.pdf
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备选作者
Tonegawa, Yoshihiro
备用出版商
Springer Science + Business Media Singapore Pte Ltd
备用出版商
Springer Nature Singapore
备用版本
SpringerBriefs in Mathematics, 1st edition 2019, Singapore, 2019
备用版本
Springer Nature, Singapore, 2019
备用版本
Singapore, Singapore
备用版本
Apr 17, 2019
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3, 20190409
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sm75027733
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Adobe PDF Library 15.0
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元数据中的注释
Source title: Brakke's Mean Curvature Flow: An Introduction (SpringerBriefs in Mathematics)
备用描述
This book explains the notion of Brakke’s mean curvature flow and its existence and regularity theories without assuming familiarity with geometric measure theory. The focus of study is a time-parameterized family of k -dimensional surfaces in the n -dimensional Euclidean space (1 ≤ k < n ). The family is the mean curvature flow if the velocity of motion of surfaces is given by the mean curvature at each point and time. It is one of the simplest and most important geometric evolution problems with a strong connection to minimal surface theory. In fact, equilibrium of mean curvature flow corresponds precisely to minimal surface. Brakke’s mean curvature flow was first introduced in 1978 as a mathematical model describing the motion of grain boundaries in an annealing pure metal. The grain boundaries move by the mean curvature flow while retaining singularities such as triple junction points. By using a notion of generalized surface called a varifold from geometric measure theory which allows the presence of singularities, Brakke successfully gave it a definition and presented its existence and regularity theories. Recently, the author provided a complete proof of Brakke’s existence and regularity theorems, which form the content of the latter half of the book. The regularity theorem is also a natural generalization of Allard’s regularity theorem, which is a fundamental regularity result for minimal surfaces and for surfaces with bounded mean curvature. By carefully presenting a minimal amount of mathematical tools, often only with intuitive explanation, this book serves as a good starting point for the study of this fascinating object as well as a comprehensive introduction to other important notions from geometric measure theory.
备用描述
This book explains the notion of Brakke's mean curvature flow and its existence and regularity theories without assuming familiarity with geometric measure theory. The focus of study is a time-parameterized family of k-dimensional surfaces in the n-dimensional Euclidean space (1 d"k <n). The family is the mean curvature flow if the velocity of motion of surfaces is given by the mean curvature at each point and time. It is one of the simplest and most important geometric evolution problems with a strong connection to minimal surface theory. In fact, equilibrium of mean curvature flow corresponds precisely to minimal surface. Brakke's mean curvature flow was first introduced in 1978 as a mathematical model describing the motion of grain boundaries in an annealing pure metal. The grain boundaries move by the mean curvature flow while retaining singularities such as triple junction points. By using a notion of generalized surface called a varifold from geometric measure theory which allows the presence of singularities, Brakke successfully gave it a definition and presented its existence and regularity theories. Recently, the author provided a complete proof of Brakke's existence and regularity theorems, which form the content of the latter half of the book. The regularity theorem is also a natural generalization of Allard's regularity theorem, which is a fundamental regularity result for minimal surfaces and for surfaces with bounded mean curvature. By carefully presenting a minimal amount of mathematical tools, often only with intuitive explanation, this book serves as a good starting point for the study of this fascinating object as well as a comprehensive introduction to other important notions from geometric measure theory
备用描述
Preface 6
Contents 9
1 Preliminary Materials 11
1.1 Basic Notation 11
1.2 Countably Rectifiable Sets 14
1.3 Varifolds 17
1.4 Mapping Varifolds 20
1.5 The First Variation of a Varifold 21
1.6 Examples 27
1.7 Some Additional Properties of Integral Varifolds 28
2 Definition of the Brakke Flow 32
2.1 Weak Formulation of Velocity 32
2.2 Weak Formulation of Normal Velocity for Varifolds and the Brakke Flow 35
2.3 Remarks on the Definition 37
2.4 The Brakke Flow in General Riemannian Manifolds 40
3 Basic Properties of the Brakke Flow 42
3.1 Continuity Property of the Brakke Flow 42
3.2 Huisken's Monotonicity Formula 43
3.3 Compactness Property for the Brakke Flow 47
3.4 Tangent Flows 53
4 A General Existence Theorem for a Brakke Flow in Codimension One 58
4.1 Main Existence Result 58
4.2 A First Try, and Its Problems 60
4.3 Open Partitions and Admissible Lipschitz Maps 63
4.4 Restricted Class of Test Functions and Area–Reducing Admissible Functions 65
4.5 Construction of Approximate Mean Curvature Flow 68
4.6 Estimates Related to h and the Measure–Reducing Property 69
4.7 Taking a Limit 74
4.8 Compactness Theorems and Last Steps 75
4.9 Comments on the Existence Results 78
5 Allard Regularity Theory 80
5.1 Time-Independent Brakke Flows 80
5.2 The Allard Regularity Theorem 82
5.3 A Glimpse at the Proof of the Allard Theorem 86
6 Regularity Theory for the Brakke Flow 94
6.1 Main Regularity Theorems 94
6.2 Outline of Proof for the Regularity Theorems 99
6.3 Comments on the Regularity Results 105
References 107
备用描述
Front Matter ....Pages i-xii
Preliminary Materials (Yoshihiro Tonegawa)....Pages 1-21
Definition of the Brakke Flow (Yoshihiro Tonegawa)....Pages 23-32
Basic Properties of the Brakke Flow (Yoshihiro Tonegawa)....Pages 33-48
A General Existence Theorem for a Brakke Flow in Codimension One (Yoshihiro Tonegawa)....Pages 49-70
Allard Regularity Theory (Yoshihiro Tonegawa)....Pages 71-84
Regularity Theory for the Brakke Flow (Yoshihiro Tonegawa)....Pages 85-97
Back Matter ....Pages 99-100
备用描述
SpringerBriefs in Mathematics
Erscheinungsdatum: 17.04.2019
开源日期
2019-04-10
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