Geometric Integration Theory

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.

Résumé
Geometric measure theory has roots going back to ancient Greek mathematics, for considerations of the isoperimetric problem (to ?nd the planar domain of given perimeter having greatest area) led naturally to questions about spatial regions and boundaries. In more modern times, the Plateau problem is considered to be the wellspring of questions in geometric measure theory. Named in honor of the nineteenth century Belgian physicist Joseph Plateau, who studied surface tension phenomena in general, andsoap?lmsandsoapbubblesinparticular,thequestion(initsoriginalformulation) was to show that a ?xed, simple, closed curve in three-space will bound a surface of the type of a disk and having minimal area. Further, one wishes to study uniqueness for this minimal surface, and also to determine its other properties. Jesse Douglas solved the original Plateau problem by considering the minimal surfacetobeaharmonicmapping(whichoneseesbystudyingtheDirichletintegral). For this work he was awarded the Fields Medal in 1936. Unfortunately, Douglass methods do not adapt well to higher dimensions, so it is desirable to ?nd other techniques with broader applicability. Enter the theory of currents. Currents are continuous linear functionals on spaces of differential forms.

Contenu

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.