Induced Fubini-Study metrics on strictly pseudoconvex CR manifolds and zeros of random CR functions
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917569703706624 |
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| author | Herrmann, Hendrik Hsiao, Chin-Yu Marinescu, George Shen, Wei-Chuan |
| author_facet | Herrmann, Hendrik Hsiao, Chin-Yu Marinescu, George Shen, Wei-Chuan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_09143 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Induced Fubini-Study metrics on strictly pseudoconvex CR manifolds and zeros of random CR functions Herrmann, Hendrik Hsiao, Chin-Yu Marinescu, George Shen, Wei-Chuan Complex Variables Analysis of PDEs Differential Geometry Probability 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. |
| title | Induced Fubini-Study metrics on strictly pseudoconvex CR manifolds and zeros of random CR functions |
| topic | Complex Variables Analysis of PDEs Differential Geometry Probability |
| url | https://arxiv.org/abs/2401.09143 |