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Self-contained, inclusive, and accessible for both the graduate students and researchers
Motivates the key ideas with examples and figures
Includes considerable background material and complete proofs
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 various 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.
Key features of Geometric Integration Theory:
* Includes topics on the deformation theorem, the area and coarea formulas, the compactness theorem, the slicing theorem and applications to minimal surfaces
* Applies techniques to complex geometry, partial differential equations, harmonic analysis, differential geometry, and many other parts of mathematics
* Provides considerable background material for the student
Motivating key ideas with examples and figures, Geometric Integration Theory is a comprehensive introduction ideal for use in the classroom and for self-study. The exposition demands minimal background, is self-contained and accessible, and thus is ideal for graduate students and researchers.
Content Level »Research
Keywords »Area formula - Plateau's problem - currents - differential forms - geometric measure theory - linear functionals - measure theory
Basics.- Carathéodory’s Construction and Lower-Dimensional Measures.- Invariant Measures and the Construction of Haar Measure..- 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.- to Currents.- Currents and the Calculus of Variations.- Regularity of Mass-Minimizing Currents.