The module of unitarily invariant area measures

TL;DR

This paper explores the hermitian analog of Aleksandrov's area measures, characterizing those arising from unitarily invariant valuations and describing the structure of smooth unitarily invariant area measures.

Contribution

It provides a characterization of area measures as first variations of unitarily invariant valuations and explicitly describes the module structure of unitarily invariant area measures.

Findings

Characterization of area measures as first variations of valuations

Smooth area measures form a module over smooth valuations

Explicit description of the module of unitarily invariant area measures

Abstract

The hermitian analog of Aleksandrov's area measures of convex bodies is investigated. A characterization of those area measures which arise as the first variation of unitarily invariant valuations is established. General smooth area measures are shown to form a module over smooth valuations and the module of unitarily invariant area measures is described explicitly.