Geometric measure theory : a beginner's guide / Frank Morgan ; illustrated by James F. Bredt.

Format:

Book

Author/Creator:

Morgan, Frank, author.

Contributor:

Bredt, James F., illustrator.

Language:

English

Subjects (All):

Geometric measure theory.

Physical Description:

1 online resource (274 p.)

Edition:

5th ed.

Place of Publication:

Amsterdam, [Netherlands] : Academic Press, 2016.

Summary:

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

Contents:

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

ExercisesChapter 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 IntegrandsExercises; 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 ν, Varifolds, and (M, ε, δ)-Minimal Sets

Notes:

Description based upon print version of record.

Includes bibliographical references and indexes.

Description based on online resource; title from PDF title page (ebrary, viewed May 18, 2016).

ISBN:

0-12-804527-2