This little book provides the newcomer or graduate student with'an illustrated introduction to geometric measure theory: the basic ideas，terminology，and results.It developed from my one-semester course at MIT for graduate students with a semester of graduate real analysis behind them.I have included a few fundamental arguments and a superficial discussion of the regularity theory，but my goal is merely to introduce the subject and make the standard text.Geometric Measure Theory by H.Federer.more accessible.

Preface

1 Geometric Measure Theory

2 Measures

3 Lipschitz Functions and Rectifiable Sets

4 Normal and Rectifiable Currents

5 The Compactness Theorem and the Existence of Area-Minimizing Surfaces

6 Examples of Area-Minimizing Surfaces

7 The Approximation Theorem

8 Survey of Regularity Results

9 Monotonicity and Oriented Tangent Cones

10 The Regularity of Area-Minimizing Hypersurfaces

11 Flat Chains Modulo v Varifolds，and-Minimal Sets

12 Miscellaneous Useful Results

13 Soap Bubble Clusters

14 Proof of Double Bubble Conjecture

15 The Hexagonal Honeycomb and Kelvin Conjectures

16 Immiscible Fluids and Crystals

17 Isoperimetric Theorems in General Codimension

18 Manifolds with Density and Perelman's Proof of the Poincare Conjecture

19 Double Bubbles in Spheres，Gauss Space，and Tori

Solutions to Exercises

Bibliography

Index of Symbols

Name Index

Subject Index