Fefferman's Hypersurface Measure and Volume Approximation Problems.

Abstract

In this thesis, we give some alternate characterizations of Fefferman's hypersurface measure on the boundary of a strongly pseudoconvex domain in complex Euclidean space. Our results exhibit a common theme: we connect Fefferman's measure to the limiting behavior of the volumes of the gap between a domain and its (suitably chosen) approximants. In one approach, these approximants are polyhedral objects with increasing complexity --- a construction inspired by similar results in convex geometry. In our second approach, the super-level sets of the Bergman kernel is the choice of approximants. In both these cases, we provide examples of some (non-strongly) pseudoconvex domains where these alternate characterizations lead to boundary measures that are invariant under volume-preserving biholomorphisms, thus extending the scope of Fefferman's original definition.