Geometric Measure Theory : A Beginner's Guide

__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

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

Title page......Page 1 Dedication......Page 3 ISBN: 978-0-12-804489-6......Page 4 Preface......Page 5 Table of contents......Page 7 1 Geometric Measure Theory......Page 9 2 Measures......Page 16 3 Lipschitz Functions and Rectifiable Sets......Page 29 4 Normal and Rectifiable Currents......Page 43 5 The Compactness Theorem and the Existence of Area-Minimizing Surfaces......Page 65 6 Examples of Area-Minimizing Surfaces......Page 73 7 The Approximation Theorem......Page 82 8 Survey of Regularity Results......Page 85 9 Monotonicity and Oriented Tangent Cones......Page 90 10 The Regularity of Area-Minimizing Hypersurfaces......Page 98 11 Flat Chains Modulo ν, Varifolds, and (M, ε, δ)-Minimal Sets......Page 106 12 Miscellaneous Useful Results......Page 112 13 Soap Bubble Clusters......Page 120 14 Proof of Double Bubble Conjecture......Page 141 15 The Hexagonal Honeycomb and Kelvin Conjectures......Page 157 16 Immiscible Fluidsand Crystals......Page 171 17 Isoperimetric Theoremsin General Codimension......Page 177 18 Manifolds with Density and Perelman’s Proof of the Poincaré Conjecture......Page 181 19 Double Bubbles in Spheres, Gauss Space, and Tori......Page 194 20 The Log-Convex Density Theorem......Page 202 Solutions to Exercises......Page 209 Bibliography......Page 231 Index of Symbols......Page 251 Name Index......Page 253 Subject Index......Page 255 On my way......Page 260