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Induced Fubini-Study metrics on strictly pseudoconvex CR manifolds and zeros of random CR functions

Hendrik Herrmann, Chin-Yu Hsiao, George Marinescu, Wei-Chuan Shen

Abstract

Let $X$ be a compact strictly pseudoconvex embeddable Cauchy-Riemann manifold and let $T_P$ 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 $χ(k^{-1}T_P)$, 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 $χ(k^{-1}λ)f_λ$, where $λ$ is an eigenvalue of $T_P$ and $f_λ$ 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.

Induced Fubini-Study metrics on strictly pseudoconvex CR manifolds and zeros of random CR functions

Abstract

Let be a compact strictly pseudoconvex embeddable Cauchy-Riemann manifold and let be the Toeplitz operator on associated with a first-order pseudodifferential operator . In our previous work we established the asymptotic expansion for large of the kernel of the operators , where is a smooth cut-off function supported in the positive real line. By using these asymptotics, we show in this paper that can be projectively embedded by maps with components of the form , where is an eigenvalue of and 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.
Paper Structure (22 sections, 66 theorems, 370 equations)

This paper contains 22 sections, 66 theorems, 370 equations.

Table of Contents

  1. Introduction and Main Results
  2. Preliminaries
  3. Notations
  4. Microlocal analysis of the Szegő kernel
  5. Semi-classical spectral asymptotics for Toeplitz operators
  6. Projective Embeddings via $T_P$
  7. Projective Embeddability
  8. Proof of Theorem \ref{['thm:ProjectiveEmbeddingIntro']}
  9. Integral Estimates for $h^{F_k}$ and $\chi_k(T_P)$
  10. Lelong-Poincaré Formula for Domains with Boundary
  11. Existence of Some Integrals
  12. Zero Sets of Holomorhic Functions on Domains with Boundary
  13. Zero Sets of CR Functions
  14. Equidistribution on CR manifolds
  15. Proof of Theorem \ref{['thm:ExpectationValueCRDistributionIntro']}
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $(X,T^{1,0}X)$ be an orientable compact strictly pseudoconvex Cauchy--Riemann manifold of dimension $2n+1$, $n\geq1$, such that the Kohn Laplacian on $X$ has closed range in $L^2(X)$. Let $\xi$ be a contact form on $X$ such that its associated Levi form $\mathcal{L}$ is positive definite. Consid where $N_k=\#\{j\mid \lambda_j\leq\delta_2k\}$ and $\chi_{k}(\lambda):=\chi\left(k^{-1}\lambda\righ

Theorems & Definitions (155)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Corollary 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Example 1.9
  • Theorem 1.10
  • ...and 145 more