Geometric measure theory and geometric function theory

Abstract

In this chapter we introduce some basic notions which are crucial for the development of sub-Riemannian geometric measure theory. Our treatment here is brief, focusing only on those aspects most relevant for the isoperimetric problem. We review and discuss Pansu’s formulation of the Rademacher differentiation theorem for Lipschitz functions on the Heisenberg group, and the basic area and co-area formulas. As an application of the former we sketch the equivalence of horizontal perimeter and Minkowski 3-content in ℍ. In Section 6.4 we present two derivations of first variation formulas for the horizontal perimeter: first, away from the characteristic locus, and second, across the characteristic locus. In the final section, we give a rough outline of Mostow’s rigidity theorem for cocompact lattices in the complex hyperbolic space H ℂ2, emphasizing the appearing of sub-Riemannian geometric function theory in the asymptotic analysis of boundary maps on the sphere at infinity.