Volume Approximations of Strictly Pseudoconvex Domains

TL;DR

This paper explores the relationship between Fefferman's hypersurface measure and the Bergman kernel in strictly pseudoconvex domains, linking complex analysis, geometry, and approximation theory.

Contribution

It introduces a novel interpretation of Fefferman's measure via Bergman kernels, extending convex geometric concepts to complex holomorphic settings.

Findings

Fefferman's measure can be recovered from the Bergman kernel.

A connection between Fefferman's measure and Heisenberg group geometry is established.

The approach provides an alternative perspective on boundary measures in complex domains.

Abstract

In convex geometry, the Blaschke surface area measure on the boundary of a convex domain can be interpreted in terms of the complexity of approximating polyhedra. In response to a question raised by D. Barrett, this approach is formulated in the holomorphic setting to establish an alternate interpretation of Fefferman's hypersurface measure on boundaries of strictly pseudoconvex domains in $\mathbb{C}^2$. In particular, it is shown that Fefferman's measure can be recovered from the Bergman kernel of the domain. A connection with the geometry of the Heisenberg group, emerging from these results, is also discussed.