Equidistribution theorems on strongly pseudoconvex domains

TL;DR

This paper proves equidistribution theorems for zeros of CR functions on strongly pseudoconvex CR manifolds and holomorphic functions on complex manifolds with strongly pseudoconvex boundaries, using kernel estimates.

Contribution

It establishes new equidistribution results for zeros of CR and holomorphic functions on strongly pseudoconvex domains, utilizing Szegő and Bergman kernel asymptotics.

Findings

Equidistribution of zeros of CR functions on strongly pseudoconvex CR manifolds.

Equidistribution of zeros of holomorphic functions on complex manifolds with pseudoconvex boundary.

Use of Szegő and Bergman kernel estimates to prove distribution results.

Abstract

This work consists of two parts. In the first part, we consider a compact connected strongly pseudoconvex CR manifold $X$ with a transversal CR $S^{1}$ action. We establish an equidistribution theorem on zeros of CR functions. The main techniques involve a uniform estimate of Szeg\H{o} kernel on $X$. In the second part, we consider a general complex manifold $M$ with a strongly pseudoconvex boundary $X$. By using classical result of Boutet de Monvel-Sj\"ostrand about Bergman kernel asymptotics, we establish an equidistribution theorem on zeros of holomorphic functions on $\overline M$.