Geometric Integration Theory. Cornerstones

This textbook introduces geometric measure theory through the notion of currents. Currents, continuous linear functionals on spaces of differential forms, are a natural language in which to formulate types of extremal problems arising in geometry, and can be used to study generalized versions of the Plateau problem and related questions in geometric analysis.
The text provides considerable background for the student and discusses techniques that are applicable to complex geometry, partial differential equations, harmonic analysis, differential geometry, and many other parts of mathematics. Topics include the deformation theorem, the area and coareas formulas, the compactness theorem, the slicing theorem and applications to minimal surfaces.
Motivating key ideas with examples and figures, Geometric Integration Theory is a comprehensive introduction ideal for both use in the classroom and for self-study. The exposition demands minimal background, is self-contained and accessible, and thus is ideal for both graduate students and researchers. TOC:Preface.- Basics.- Carathéodory's Construction and Lower-Dimensional Measures.- Invariant Meaures and the Construction of Haar Meaure.- Covering Theorems and the Differentiation of Integrals.- Analytical Tools: the Area Formula, the Coarea Formula, and Poincaré Inequalities.- The Calculus of Differential Forms and Stokes's Theorem.- Introduction to Currents.- Currents and the Calculus of Variations.- Regularity of Mass-Minimizing Currents.- Appendix.-References.- Index of Notation.- Index.

This textbook introduces geometric measure theory through the notion of currents. Currents, continuous linear functionals on spaces of differential forms, are a natural language in which to formulate types of extremal problems arising in geometry, and can be used to study generalized versions of the Plateau problem and related questions in geometric analysis.
The text provides considerable background for the student and discusses techniques that are applicable to complex geometry, partial differential equations, harmonic analysis, differential geometry, and many other parts of mathematics. Topics include the deformation theorem, the area and coareas formulas, the compactness theorem, the slicing theorem and applications to minimal surfaces.
Motivating key ideas with examples and figures, Geometric Integration Theory is a comprehensive introduction ideal for both use in the classroom and for self-study. The exposition demands minimal background, is self-contained and accessible, and thus is ideal for both graduate students and researchers.

Preface.- Basics.- Carathéodory’s Construction and Lower-Dimensional Measures.- Invariant Meaures and the Construction of Haar Meaure.- Covering Theorems and the Differentiation of Integrals.- Analytical Tools: the Area Formula, the Coarea Formula, and Poincaré Inequalities.- The Calculus of Differential Forms and Stokes’s Theorem.- Introduction to Currents.- Currents and the Calculus of Variations.- Regularity of Mass-Minimizing Currents.- Appendix.-References.- Index of Notation.- Index.