CHAPTER 1 - Geometric Measure Theory

Abstract

Geometric measure theory could be described as differential geometry, generalized through measure theory to deal with maps and surfaces that are not necessarily smooth and applied to the calculus of variations. It dates from the 1960 foundational paper of Herbert Federer and Wendell Fleming on Normal and Integral Currents, recognized by the 1986 AMS Steele Prize for a paper of fundamental or lasting importance. This chapter presents an outline of the purpose and basic concepts of geometric measure theory. Along with its successes and advantages, the definition of a surface as a mapping has certain drawbacks: (1) there is an inevitable a priori restriction on the types of singularities that can occur; (2) there is an a priori restriction on the topological complexity; and (3) the natural topology lacks compactness properties. The importance of compactness properties appears in the direct method. The direct method for finding a surface of least area with a given boundary has three steps: (1) take a sequence of surfaces with areas decreasing to the infimum, (2) extract a convergent subsequence, and (3) show that the limit surface is the desired surface of least area. An alternative to surfaces as mappings is provided by rectifiable currents, the m-dimensional, oriented surfaces of geometric measure theory.