Geometric Measure Theory

A Beginner's Guide

Frank Morgan

• Publisher: Academic Press
• Year: 2016
• ISBN:
  9780128045275
• Edition: 5
• Topic: Matemáticas
• Subtopic: Geometría

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

Part I

Basic Theory

1

Geometric Measure Theory

Geometric measure theory could be described as differential geometry, generalized through measure theory to deal with maps and surfaces that are not necessarily smooth, and applied to the calculus of variations. It dates from the 1960 foundational paper of Herbert Federer and Wendell Fleming on “Normal and Integral Currents,” recognized by the 1986 AMS Steele Prize for a paper of fundamental or lasting importance, and earlier and contemporaneous work of L. C. Young [1, 2], E. De Giorgi [1, 3, 4], and E. R. Reifenberg [1–3] (see Figure 1.0.1). This chapter provides a rough outline of the purpose and basic concepts of geometric measure theory. Later chapters take up these topics more carefully.

Figure 1.0.1 Wendell Fleming, Fred Almgren, and Ennio De Giorgi, three of the founders of geometric measure theory, at the Scuola Normale Superiore, Pisa, summer, 1965; and Fleming today.

Photographs courtesy of Fleming.

1.1 Archetypical Problem

Given a boundary in Rⁿ, find the surface of least area with that boundary. See Figure 1.1.1. Progress on this problem depends crucial...