Geometric Measure Theory

A Beginner's Guide
 
 
Academic Press
  • 5. Auflage
  • |
  • erschienen am 2. Mai 2016
  • |
  • 272 Seiten
 
E-Book | ePUB mit Adobe DRM | Systemvoraussetzungen
E-Book | PDF mit Adobe DRM | Systemvoraussetzungen
978-0-12-804527-5 (ISBN)
 

Geometric Measure Theory: A Beginner's Guide, Fifth Edition provides the framework readers need to understand the structure of a crystal, a soap bubble cluster, or a universe.

The book is essential to any student who wants to learn geometric measure theory, and will appeal to researchers and mathematicians working in the field. Brevity, clarity, and scope make this classic book an excellent introduction to more complex ideas from geometric measure theory and the calculus of variations for beginning graduate students and researchers.

Morgan emphasizes geometry over proofs and technicalities, providing a fast and efficient insight into many aspects of the subject, with new coverage to this edition including topical coverage of the Log Convex Density Conjecture, a major new theorem at the center of an area of mathematics that has exploded since its appearance in Perelman's proof of the Poincaré conjecture, and new topical coverage of manifolds taking into account all recent research advances in theory and applications.


  • Focuses on core geometry rather than proofs, paving the way to fast and efficient insight into an extremely complex topic in geometric structures
  • Enables further study of more advanced topics and texts
  • Demonstrates in the simplest possible way how to relate concepts of geometric analysis by way of algebraic or topological techniques
  • Contains full topical coverage of The Log-Convex Density Conjecture
  • Comprehensively updated throughout


Frank Morgan is the Dennis Meenan '54 Third Century Professor of Mathematics at Williams College. He obtained his B.S. from MIT and his M.S. and Ph.D. from Princeton University. His research interest lies in minimal surfaces, studying the behavior and structure of minimizers in various settings. He has also written Riemannian Geometry: A Beginner's Guide, Calculus Lite, and most recently The Math Chat Book, based on his television program and column on the Mathematical Association of America Web site.
  • Englisch
  • San Diego
  • |
  • USA
Elsevier Science
  • 50,27 MB
978-0-12-804527-5 (9780128045275)
0128045272 (0128045272)
weitere Ausgaben werden ermittelt
  • Front Cover
  • Dedication
  • Geometric Measure Theory: A Beginner's Guide
  • Copyright
  • Contents
  • Preface
  • Part I: Basic Theory
  • Chapter 1: Geometric Measure Theory
  • 1.1 Archetypical Problem
  • 1.2 Surfaces as Mappings
  • 1.3 The Direct Method
  • 1.4 Rectifiable Currents
  • 1.5 The Compactness Theorem
  • 1.6 Advantages of Rectifiable Currents
  • 1.7 The Regularity of Area-Minimizing Rectifiable Currents
  • 1.8 More General Ambient Spaces
  • Chapter 2: Measures
  • 2.1 Definitions
  • 2.2 Lebesgue Measure
  • 2.3 Hausdorff Measure
  • 2.4 Integral-Geometric Measure
  • 2.5 Densities
  • 2.6 Approximate Limits
  • 2.7 Besicovitch Covering Theorem
  • 2.8 Corollary
  • 2.9 Corollary
  • 2.10 Corollary
  • Exercises
  • Chapter 3: Lipschitz Functions and Rectifiable Sets
  • 3.1 Lipschitz Functions
  • 3.2 Rademacher's Theorem
  • 3.3 Approximation of a Lipschitz Function by a C1 Funcation
  • 3.4 Lemma (Whitney's Extension Theorem)
  • 3.5 Proposition
  • 3.6 Jacobians
  • 3.7 The Area Formula
  • 3.8 The Coarea Formula
  • 3.9 Tangent Cones
  • 3.10 Rectifiable Sets
  • 3.11 Proposition
  • 3.12 Proposition
  • 3.13 General Area-Coarea Formula
  • 3.14 Product of Measures
  • 3.15 Orientation
  • 3.16 Crofton's Formula
  • 3.17 Structure Theorem
  • Exercises
  • Chapter 4: Normal and Rectifiable Currents
  • 4.1 Vectors and Differential Forms
  • 4.2 Currents
  • 4.3 Important Spaces of Currents
  • 4.3A Mapping Currents
  • 4.3B Currents Representable by Integration
  • 4.4 Theorem
  • 4.5 Normal Currents
  • 4.6 Proposition
  • 4.7 Theorem
  • 4.8 Theorem
  • 4.9 Constancy Theorem
  • 4.10 Cartesian Products
  • 4.11 Slicing
  • 4.12 Lemma
  • 4.13 Proposition
  • Exercises
  • Chapter 5: The Compactness Theorem and the Existence of Area-Minimizing Surfaces
  • 5.1 The Deformation Theorem
  • 5.2 Corollary
  • 5.3 The Isoperimetric Inequality
  • 5.4 The Closure Theorem
  • 5.5 The Compactness Theorem
  • 5.6 The Existence of Area-Minimizing Surfaces
  • 5.7 The Existence of Absolutely and Homologically Minimizing Surfaces in Manifolds
  • Exercises
  • Chapter 6: Examples of Area-Minimizing Surfaces
  • 6.1 The Minimal Surface Equation
  • 6.2 Remarks on Higher Dimensions
  • 6.3 Complex Analytic Varieties
  • 6.4 Fundamental Theorem of Calibrations
  • 6.5 History of Calibrations
  • Exercises
  • Chapter 7: The Approximation Theorem
  • 7.1 The Approximation Theorem
  • Chapter 8: Survey of Regularity Results
  • 8.1 Theorem
  • 8.2 Theorem
  • 8.3 Theorem
  • 8.4 Boundary Regularity
  • 8.5 General Ambients, Volume Constraints, and Other Integrands
  • Exercises
  • Chapter 9: Monotonicity and Oriented Tangent Cones
  • 9.1 Locally Integral Flat Chains
  • 9.2 Monotonicity of the Mass Ratio
  • 9.3 Theorem
  • 9.4 Corollary
  • 9.5 Corollary
  • 9.6 Corollary
  • 9.7 Oriented Tangent Cones
  • 9.8 Theorem
  • 9.9 Theorem
  • Exercises
  • Chapter 10: The Regularity of Area-Minimizing Hypersurfaces
  • 10.1 Theorem
  • 10.2 Regularity for Area-Minimizing Hypersurfaces Theorem
  • 10.3 Lemma
  • 10.4 Maximum Principle
  • 10.5 Simons's Lemma
  • 10.6 Lemma
  • 10.7 Remarks
  • Exercises
  • Chapter 11: Flat Chains Modulo ?, Varifolds, and (M, e, d)-Minimal Sets
  • 11.1 Flat Chains Modulo ?
  • 11.2 Varifolds
  • 11.3 (M, e, d)-Minimal Sets
  • Exercises
  • Chapter 12: Miscellaneous Useful Results
  • 12.1 Morse-Sard-Federer Theorem
  • 12.2 Gauss-Green-De Giorgi-Federer Theorem
  • 12.3 Relative Homology
  • 12.4 Functions of Bounded Variation
  • 12.5 General Parametric Integrands
  • Part II: Applications
  • Chapter 13: Soap Bubble Clusters
  • 13.1 Planar BubbleClusters
  • 13.1A Theorem
  • 13.1B Connected Regions
  • 13.2 Theory of Single Bubbles
  • 13.3 Cluster Theory
  • 13.4 Existence of Soap Bubble Clusters
  • 13.5 Lemma
  • 13.6 Lemma
  • 13.7 Sketch of Proof of Theorem 13.4
  • 13.8 Proposition
  • 13.9 Regularity of Soap Bubble Clusters in R3
  • 13.10 Cluster Regularity in Higher Dimensions
  • 13.11 Minimizing Surface and Curve Energies
  • 13.12 Bubble Cluster Equilibrium
  • 13.13 Von Neumann's Law
  • 13.14 Helmholtz Free Energy
  • Exercises
  • Chapter 14: Proof of Double Bubble Conjecture
  • 14.1 Proposition
  • 14.2 Remark
  • 14.3 Theorem
  • 14.4 Lemma
  • 14.5 Concavity
  • 14.6 Corollary
  • 14.7 Corollary
  • 14.8 Decomposition Lemma
  • 14.9 Hutchings Basic Estimate
  • 14.10 Hutchings Structure Theorem
  • 14.11 Corollary
  • 14.12 Renormalization
  • 14.13 Remark on Rigor
  • 14.14 Proposition
  • 14.15 Proposition
  • 14.16 Corollary
  • 14.17 Corollary
  • 14.18 Proposition
  • 14.19 Proposition
  • 14.20 Theorem
  • 14.21 Open Questions
  • 14.22 Physical Stability
  • Exercises
  • Chapter 15: The Hexagonal Honeycomb and Kelvin Conjectures
  • 15.1 Hexagonal Honeycomb History
  • 15.2 Definition of Clusters in R2
  • 15.3 The Truncation Lemma
  • 15.4 Hexagonal Isoperimetric Inequality
  • 15.5 Chordal Isoperimetric Inequality
  • 15.6 Proposition
  • 15.7 The Hexagonal Honeycomb Theorem
  • 15.8 The Bees' Honeycomb
  • 15.9 Unequal Areas
  • 15.10 Kelvin Conjecture Disproved by Weaire and Phelan
  • 15.11 Higher Dimensions
  • 15.12 How the Weaire-Phelan Counterexample to the Kelvin Conjecture Could Have Been Found Earlier
  • 15.13 Conjectures and Proofs
  • Exercise
  • Chapter 16: Immiscible Fluids and Crystals
  • 16.1 Immiscible Fluids
  • 16.2 Existence of Minimizing Fluid Clusters
  • 16.3 Regularity of Minimizing Fluid Clusters
  • 16.4 Crystals
  • 16.5 Planar Double Crystals of Salt
  • 16.6 Willmore and Knot Energies
  • 16.7 Flows and Crystal Growth
  • 16.8 The Brakke Evolver
  • Chapter 17: Isoperimetric Theorems in General Codimension
  • 17.1 Theorem
  • 17.2 Theorem
  • 17.3 General Ambient Manifolds
  • Chapter 18: Manifolds with Density and Perelman's Proof of the Poincaré Conjecture
  • 18.1 Definitions
  • 18.2 Theorem
  • 18.3 Curvature
  • 18.4 Theorem (Classical Heintze-Karcher)
  • 18.5 Theorem (Heintze-Karcher for Manifolds with Density)
  • 18.6 Theorem (Classical Levy-Gromov)
  • 18.7 Theorem (Levy-Gromov for Manifolds with Density)
  • 18.8 Myers' Theorem with Density
  • 18.9 First and Second Variation
  • 18.10 Corollary
  • 18.11 Perelman's Proof of the Poincaré Conjecture
  • Exercises
  • Chapter 19: Double Bubbles in Spheres, Gauss Space, and Tori
  • 19.1 Double Bubbles in Spheres
  • 19.2 Component Bound
  • 19.3 Instability Argument
  • 19.4 Double Bubble Theorems in Sn
  • 19.5 Double Bubble Theorem in Gn
  • 19.6 Double Bubbles in Flat Two-Tori
  • 19.7 The Cubic Three-Torus
  • Exercises
  • Chapter 20: The Log-Convex Density Theorem
  • 20.1 Proposition
  • 20.2 Remarks
  • 20.3 Corollary
  • 20.4 Existence and regularity
  • 20.5 History of the Log-Convex Density Conjecture
  • 20.6 The Log-Convex Density Theorem
  • 20.7 Theorem
  • Exercises
  • Solutions to Exercises
  • Chapter 2
  • Chapter 3
  • Chapter 4
  • Chapter 5
  • Chapter 6
  • Chapter 8
  • Chapter 9
  • Chapter 10
  • Chapter 11
  • Chapter 13
  • Chapter 14
  • Chapter 15
  • Chapter 18
  • Chapter 19
  • Chapter 20
  • Bibliography
  • Index of Symbols
  • Name Index
  • Subject Index
  • On my way
  • Back Cover

Dateiformat: EPUB
Kopierschutz: Adobe-DRM (Digital Rights Management)

Systemvoraussetzungen:

Computer (Windows; MacOS X; Linux): Installieren Sie bereits vor dem Download die kostenlose Software Adobe Digital Editions (siehe E-Book Hilfe).

Tablet/Smartphone (Android; iOS): Installieren Sie bereits vor dem Download die kostenlose App Adobe Digital Editions (siehe E-Book Hilfe).

E-Book-Reader: Bookeen, Kobo, Pocketbook, Sony, Tolino u.v.a.m. (nicht Kindle)

Das Dateiformat EPUB ist sehr gut für Romane und Sachbücher geeignet - also für "fließenden" Text ohne komplexes Layout. Bei E-Readern oder Smartphones passt sich der Zeilen- und Seitenumbruch automatisch den kleinen Displays an. Mit Adobe-DRM wird hier ein "harter" Kopierschutz verwendet. Wenn die notwendigen Voraussetzungen nicht vorliegen, können Sie das E-Book leider nicht öffnen. Daher müssen Sie bereits vor dem Download Ihre Lese-Hardware vorbereiten.

Weitere Informationen finden Sie in unserer E-Book Hilfe.


Dateiformat: PDF
Kopierschutz: Adobe-DRM (Digital Rights Management)

Systemvoraussetzungen:

Computer (Windows; MacOS X; Linux): Installieren Sie bereits vor dem Download die kostenlose Software Adobe Digital Editions (siehe E-Book Hilfe).

Tablet/Smartphone (Android; iOS): Installieren Sie bereits vor dem Download die kostenlose App Adobe Digital Editions (siehe E-Book Hilfe).

E-Book-Reader: Bookeen, Kobo, Pocketbook, Sony, Tolino u.v.a.m. (nicht Kindle)

Das Dateiformat PDF zeigt auf jeder Hardware eine Buchseite stets identisch an. Daher ist eine PDF auch für ein komplexes Layout geeignet, wie es bei Lehr- und Fachbüchern verwendet wird (Bilder, Tabellen, Spalten, Fußnoten). Bei kleinen Displays von E-Readern oder Smartphones sind PDF leider eher nervig, weil zu viel Scrollen notwendig ist. Mit Adobe-DRM wird hier ein "harter" Kopierschutz verwendet. Wenn die notwendigen Voraussetzungen nicht vorliegen, können Sie das E-Book leider nicht öffnen. Daher müssen Sie bereits vor dem Download Ihre Lese-Hardware vorbereiten.

Weitere Informationen finden Sie in unserer E-Book Hilfe.


Download (sofort verfügbar)

85,62 €
inkl. 19% MwSt.
Download / Einzel-Lizenz
ePUB mit Adobe DRM
siehe Systemvoraussetzungen
PDF mit Adobe DRM
siehe Systemvoraussetzungen
Hinweis: Die Auswahl des von Ihnen gewünschten Dateiformats und des Kopierschutzes erfolgt erst im System des E-Book Anbieters
E-Book bestellen

Unsere Web-Seiten verwenden Cookies. Mit der Nutzung dieser Web-Seiten erklären Sie sich damit einverstanden. Mehr Informationen finden Sie in unserem Datenschutzhinweis. Ok