Geometric Integration Theory [Hardcover]

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.Basics.- Carath?odorys 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 Poincar? Inequalities..- The Calculus of Differential Forms and Stokess Theorem.- to Currents.- Currents and the Calculus of Variations.- Regularity of Mass-Minimizing Currents.

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This is a graduate textbook with the main purpose of introducing geometric measure theory through the notion of currents. & One of the most important features of this text is that it is self-contained & . The book also contains an Appendix & as well asl