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Fleeing the vengeance of her brother, Dido lands on the coast of North Africa and founds the city of Carthage. Within the mythology associated with Virgil’s saga lies one of the earliest problems in extremal geometric analysis. For the bargain which Dido agrees to with a local potentate is this: she may have that portion of land which she is able to enclose with the hide of a bull. Legend records Dido’s ingenious and elegant solution: cutting the hide into a series of long thin strips, she marks out a vast circumference, forming the eventual line of the walls of ancient Carthage. This problem is a variant of what has become known as the classical isoperimetric problem .^1 In more precise terms it may be formulated as follows: among all bounded, connected open regions in the plane with a fixed perimeter, characterize those regions with the maximal volume . Needless to say, Dido’s solution is correct: the extremal regions are precisely open circular planar discs.

In this chapter we provide a detailed description of the sub-Riemannian geometry of the first Heisenberg group. We describe its algebraic structure, introduce the horizontal subbundle (which we think of as constraints ) and present the Carnot-Carathéodory metric as the least time required to travel between two given points at unit speed along horizontal paths. Subsequently we introduce the notion of sub-Riemannian metric and show how it arises from degenerating families of Riemannian metrics. For use in later chapters we compute some of the standard differential geometric apparatus in these Riemannian approximants.

A very intuitive way to think of the sub-Riemannian Heisenberg group is as a medium in which motion is only possible along a given set of directions, changing from point to point. If the constraints are too tight, then it may be impossible to join any two points with an admissible trajectory, hence one needs to find conditions on the constraints implying “horizontal accessibility”.

This chapter is devoted to the study of the sub-Riemannian geometry of codimension 1 smooth submanifolds of the Heisenberg group.

In this chapter we review the definitions of Sobolev spaces, BV functions and perimeter of a set relative to the sub-Riemannian structure of ℍ. These notions 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.

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 _ℂ ^2 , emphasizing the appearing of sub-Riemannian geometric function theory in the asymptotic analysis of boundary maps on the sphere at infinity.

The isoperimetric inequality in ℍ with respect to the horizontal perimeter was first proved by Pansu. We first state it in the setting of C ^1 sets.

This chapter is the core of this survey. We recall the definition of isoperimetric profile of ℍ and Pansu’s 1982 conjecture. Next we present a proof of the existence of an isoperimetric profile and describe some of the existing literature on the isoperimetric problem. Our aim is to reveal the main ideas and outlines of the proofs of various partial results and sketch some further techniques and methods which may lead to a solution, in order to guide the reader through the literature and to give a sense of the larger ideas that are in play.

As the point of departure for this final chapter, we return to the equivalence of the isoperimetric inequality with the geometric ( L ^1-) Sobolev inequality. As shown in Section 7.1, the best constant for the isoperimetric inequality agrees with the best constant for the geometric ( L ^1-) Sobolev inequality. Recall that in the context of the Heisenberg group, the L ^p-Sobolev inequalities take the form 9.1 $$ \left\| u \right\|_{4p/(4 - p)} \leqslant Cp(\mathbb{H})\left\| {\nabla _0 u} \right\|_p , u \in C_0^\infty (\mathbb{H}). $$ In this chapter we discuss sharp constants for other analytic/geometric inequalities in the Heisenberg group and the Grushin plane. These include the L ^p-Sobolev inequality (9.1) in the case p = 2, the Trudinger inequality (9.14), which serves as a natural substitute for (9.1) in the limiting case p = 4, and the Hardy inequality (9.24), a weighted inequality of Sobolev type on the domain ℍ \ { o }.