Proper holomorphic mappings between generalized Hartogs triangles

Zapałowski, Paweł

doi:10.1007/s10231-016-0607-2

Cited by 7 publications

(4 citation statements)

References 17 publications

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“…Indeed, the map (z 1 , z 2 ) → (z 1 z n−1 2 , z m 2 ) is a proper map from H 1 onto H m/n , so Bell's formula gives B m/n as a finite sum. In [26], Zapalowski characterizes the proper maps between fat Hartogs triangles. He shows there is a proper map F : H m/n → H p/q if and only if there are a, b ∈ Z + such that…”

Section: Bell's Transformation Rule and Derivation Of The Kernelmentioning

confidence: 99%

“…In [9], Chakrabarti and Zeytuncu study the L p -mapping properties of the Bergman projection on the classical Hartogs triangle, and in [10], Chen studies L p -mapping of the Bergman projection on analogous domains in higher dimensions. Zapalowski, [26], characterizes proper maps between generalizations of the Hartogs triangle in C n . This author and McNeal investigate the Bergman projection on fat Hartogs triangles in [13].…”

Section: Introductionmentioning

confidence: 99%

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Bergman theory of certain generalized Hartogs triangles

Edholm¹

2016

Pacific J. Math.

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The Bergman theory of domains $\{ |{z_{1} |^{\gamma}} < |{z_{2}} | < 1 \}$ in $\mathbb{C}^2$ is studied for certain values of $\gamma$, including all positive integers. For such $\gamma$, we obtain a closed form expression for the Bergman kernel, $\mathbb{B}_{\gamma}$. With these formulas, we make new observations relating to the Lu Qi-Keng problem and analyze the boundary behavior of $\mathbb{B}_{\gamma}(z,z)$.Comment: 13 page

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Section: Bell's Transformation Rule and Derivation Of The Kernelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Bergman theory of certain generalized Hartogs triangles

Edholm¹

2016

Pacific J. Math.

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“…Recently, Zapałowski [35] gave the rigidity of proper holomorphic self-mappings between generalized Hartogs triangle and obtained automorphism group of the generalized Hartogs triangle. The reader is also referred to [11][12][13][14]23] for the studies of rigidity of the proper holomorphic mappings between Hartogs triangles.…”

Section: Introductionmentioning

confidence: 99%

Balanced metrics and Berezin quantization on Hartogs triangles

2020

Annali di Matematica

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In this paper, we study balanced metrics and Berezin quantization on a class of Hartogs domains defined by $$\varOmega _n=\{(z_1,\ldots ,z_n)\in {\mathbb {C}}^n:\vert z_1\vert<\vert z_2\vert<\cdots<\vert z_n\vert <1\}$$ Ω n = { ( z 1 , … , z n ) ∈ C n : | z 1 | < | z 2 | < ⋯ < | z n | < 1 } which generalize the so-called classical Hartogs triangle. We introduce a Kähler metric $$g(\nu )$$ g ( ν ) associated with the Kähler potential $$\varPhi _n(z):=-\sum _{k=1}^{n-1}\nu _k\ln (\vert z_{k+1}\vert ^2-\vert z_k\vert ^2)-\nu _n\ln (1-\vert z_n\vert ^2)$$ Φ n ( z ) : = - ∑ k = 1 n - 1 ν k ln ( | z k + 1 | 2 - | z k | 2 ) - ν n ln ( 1 - | z n | 2 ) on $$\varOmega _n$$ Ω n . As main contributions, on one hand we compute the explicit form for Bergman kernel of weighted Hilbert space, and then, we obtain the necessary and sufficient condition for the metric $$g(\nu )$$ g ( ν ) on the domain $$\varOmega _n$$ Ω n to be a balanced metric. On the other hand, by using the Calabi’s diastasis function, we prove that the Hartogs triangles admit a Berezin quantization.

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“…Alexander's theorem has been generalized to many classes of domains (e.g., see Bedford-Bell [3], Diederich-Fornaess [5], Huang [9], Su-Tu-Wang [17], Tu [19], Tu-Wang [20,21], and Webster [22]). Inspired by these theorems, there are many results on classifying proper holomorphic mappings up to holomorphic automorphisms (e.g., see Dini-Primicerio [6], Ebenfelt-Son [7], Faran [8], Landucci-Pinchuk [11], Spiro [16], and Zapalowski [23]).…”

Section: Introductionmentioning

confidence: 99%

Classification of Proper Holomorphic Mappings Between Certain Unbounded Non-hyperbolic Domains

Wang

2018

J Geom Anal

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The Fock-Bargmann-Hartogs domain Dn,m(µ) (µ > 0) in C n+m is defined by the inequality w 2 < e −µ z 2 , where (z, w) ∈ C n × C m , which is an unbounded non-hyperbolic domain in C n+m . Recently, Tu-Wang obtained the rigidity result that proper holomorphic self-mappings of Dn,m(µ) are automorphisms for m ≥ 2, and found a counter-example to show that the rigidity result isn't true for Dn,1(µ). In this article, we obtain a classification of proper holomorphic mappings between Dn,1(µ) and DN,1(µ) with N < 2n.. Alexander [1, 2] further studied proper holomorphic mappings between bounded domains with the same dimension. Alexander's theorem has been generalized to many classes of domains (e.g., see Bedford-Bell [3], Diederich-Fornaess [5], Huang [9], Su-Tu-Wang [17], Tu [19], Tu-Wang [20, 21], and Webster [22]). Inspired by these theorems, there are many results on classifying proper holomorphic mappings up to holomorphic automorphisms (e.g., see Dini-Primicerio [6], Ebenfelt-Son [7], Faran [8], Landucci-Pinchuk [11], Spiro [16], and Zapalowski [23]).The Fock-Bargmann-Hartogs domains D n,m (µ) are defined by D n,m (µ) := {(z, w) ∈ C n × C m : w 2 < e −µ z 2 }, µ > 0.

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Proper holomorphic mappings between generalized Hartogs triangles

Cited by 7 publications

References 17 publications

Bergman theory of certain generalized Hartogs triangles

Bergman theory of certain generalized Hartogs triangles

Balanced metrics and Berezin quantization on Hartogs triangles

Classification of Proper Holomorphic Mappings Between Certain Unbounded Non-hyperbolic Domains

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