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.


Heisenberg Group Carnot Group Constant Mean Curvature Isoperimetric Problem Minkowski Content 
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© Birkhäuser Verlag AG 2007

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