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lgli/N:\!genesis_files_for_add\_add\kolxo3\95\M_Mathematics\MD_Geometry and topology\MDdg_Differential geometry\Tonegawa Y. Brakke's mean curvature flow.. an introduction (SpringerBriefs, Springer, 2019)(ISBN 9789811370748)(O)(108s)_MDdg_.pdf
Brakke's Mean Curvature Flow: An Introduction (SpringerBriefs in Mathematics) 🔍
Tonegawa, Yoshihiro Springer Singapore : Imprint: Springer, SpringerBriefs in Mathematics, 1st edition 2019, Singapore, 2019
英语 [en] · PDF · 1.0MB · 2019 · 📘 非小说类图书 · 🚀/lgli/lgrs/nexusstc/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.
备用文件名
lgrsnf/N:\!genesis_files_for_add\_add\kolxo3\95\M_Mathematics\MD_Geometry and topology\MDdg_Differential geometry\Tonegawa Y. Brakke's mean curvature flow.. an introduction (SpringerBriefs, Springer, 2019)(ISBN 9789811370748)(O)(108s)_MDdg_.pdf
备用文件名
lgli/M_Mathematics/MD_Geometry and topology/MDdg_Differential geometry/Tonegawa Y. Brakke's mean curvature flow.. an introduction (SpringerBriefs, Springer, 2019)(ISBN 9789811370748)(O)(108s)_MDdg_.pdf
备用文件名
nexusstc/Brakke's mean curvature flow: an introduction/a27be8d8cfb802107fb3eb9b68b43f04.pdf
备用文件名
zlib/Mathematics/Tonegawa Y/Brakke's mean curvature flow: an introduction_6041715.pdf
备选作者
Yoshihiro Tonegawa
备用出版商
Springer Science + Business Media Singapore Pte Ltd
备用出版商
Springer Nature Singapore
备用版本
Springer Nature, Singapore, 2019
备用版本
Singapore, Singapore
备用版本
Apr 17, 2019
备用版本
3, 20190409
元数据中的注释
kolxo3 -- 95
元数据中的注释
lg2806455
元数据中的注释
{"isbns":["9789811370748","9789811370755","9811370745","9811370753"],"last_page":108,"publisher":"Springer","series":"SpringerBriefs"}
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 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......Page 6
Contents......Page 9
1.1 Basic Notation......Page 11
1.2 Countably Rectifiable Sets......Page 14
1.3 Varifolds......Page 17
1.4 Mapping Varifolds......Page 20
1.5 The First Variation of a Varifold......Page 21
1.6 Examples......Page 27
1.7 Some Additional Properties of Integral Varifolds......Page 28
2.1 Weak Formulation of Velocity......Page 32
2.2 Weak Formulation of Normal Velocity for Varifolds and the Brakke Flow......Page 35
2.3 Remarks on the Definition......Page 37
2.4 The Brakke Flow in General Riemannian Manifolds......Page 40
3.1 Continuity Property of the Brakke Flow......Page 42
3.2 Huisken's Monotonicity Formula......Page 43
3.3 Compactness Property for the Brakke Flow......Page 47
3.4 Tangent Flows......Page 53
4.1 Main Existence Result......Page 58
4.2 A First Try, and Its Problems......Page 60
4.3 Open Partitions and Admissible Lipschitz Maps......Page 63
4.4 Restricted Class of Test Functions and Area–Reducing Admissible Functions......Page 65
4.5 Construction of Approximate Mean Curvature Flow......Page 68
4.6 Estimates Related to h and the Measure–Reducing Property......Page 69
4.7 Taking a Limit......Page 74
4.8 Compactness Theorems and Last Steps......Page 75
4.9 Comments on the Existence Results......Page 78
5.1 Time-Independent Brakke Flows......Page 80
5.2 The Allard Regularity Theorem......Page 82
5.3 A Glimpse at the Proof of the Allard Theorem......Page 86
6.1 Main Regularity Theorems......Page 94
6.2 Outline of Proof for the Regularity Theorems......Page 99
6.3 Comments on the Regularity Results......Page 105
References......Page 107
备用描述
SpringerBriefs in Mathematics
Erscheinungsdatum: 17.04.2019
开源日期
2020-10-11
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