2024/01/17 by Herrmann, Hendrik, Chin-Yu Hsiao, Hsiao, Chin-Yu +4 · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #Analytic and geometric function theory #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Holomorphic and Operator Theory #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2401.09143
openalex publication_date 2024/01/17 · openalex created_date 2024/01/19 · openalex updated_date 2026/07/28
Let X be a compact strictly pseudoconvex embeddable Cauchy-Riemann manifold and let TP be the Toeplitz operator on X associated with a first-order pseudodifferential operator P. In our previous work we established the asymptotic expansion for k large of the kernel of the operators χ(k-1TP), where χ is a smooth cut-off function supported in the positive real line. By using these asymptotics, we show in this paper that X can be projectively embedded by maps with components of the form χ(k-1λ)fλ, where λ is an eigenvalue of TP and fλ is a corresponding eigenfunction. We establish the asymptotics of the pull-back of the Fubini-Study metric by these maps and we obtain the distribution of the zero divisors of random Cauchy-Riemann functions. We then establish a version of the Lelong-Poincaré formula for domains with boundary and obtain the distribution of the zero divisors of random holomorphic functions on strictly pseudoconvex domains.