Regular Hypersurfaces, Intrinsic Perimeter and Implicit Function Theorem in Carnot Groups

In the last few years, a systematic attempt to develop geometric measure theory in metric spaces has become the object of many studies. Such a program, already suggested in Federer's book [17], has been explicitly formulated and carried on by several authors. We only mention some of them: De Giorgi [14], [15], [16], Gromov [28], [29], Preiss and Tisĕr [44], Kirchheim [33], David & Semmes [11], Cheeger [9] and Ambrosio and Kirchheim [3], [4].

In this paper we study, inside a special class of metric spaces i.e. the Carnot groups, a classical problem in Geometric Measure Theory that is the problem of defining regular hypersurfaces and different reasonable surface measures on them, and of understanding their relationships (here hypersurface means simply codimension 1 surface).

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Published 1 January 2003