Spiralshapelike mappings in several complex variables
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Spiralshapelike mappings in several complex variables Hidetaka Hamada1 · Mihai Iancu2 · Gabriela Kohr2 Received: 3 May 2019 / Accepted: 15 February 2020 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Let n ≥ 2 and let A ∈ L(ℂn ) be such that k+ (A) < 2m(A) . In this paper, we prove that if F ∶ 𝔹n → ℂn is a normalized biholomorphic mapping such that F(𝔹n ) is a bounded strictly pseudoconvex domain with C2 boundary, then F(𝔹n ) is polynomially convex if and only if there exists a normalized automorphism 𝛷 ∶ ℂn → ℂn such that 𝛷(F(𝔹n )) is an A-spirallike domain, i.e., e−tA 𝛷(F(𝔹n )) ⊆ 𝛷(F(𝔹n )) , t ≥ 0 . Moreover, for the necessary condition, we can choose 𝛷 such that e−tA 𝛷(F(𝔹n )) ⊆ 𝛷(F(𝔹n )) , t > 0 . In particular, this result holds for A = In . This result extends a recent result due to Arosio, Bracci and Wold in the case of convexshapelike domains. Certain consequences, examples and questions are also mentioned. Keywords Convexshapelike · Filtering Loewner chain · Polynomially convex · Spiralshapelike · Starshapelike Mathematics Subject Classification 32H02 · 30C35 · 30C55 · 30C80
1 Introduction Arosio, Bracci and Wold [5] used the well-known result of Andersén and Lempert, concerning the approximation of biholomorphic mappings on starlike domains in ℂn with Runge images by automorphisms of ℂn ( n ≥ 2 ) (see [3, Theorem 2.1]; see also [14, Chapter 4]), to study a property of starshapelike domains in ℂn and the Loewner differential equation on starlike domains in ℂn . The first named author of this paper extended the work
* Hidetaka Hamada [email protected]‑u.ac.jp Mihai Iancu [email protected] Gabriela Kohr [email protected] 1
Faculty of Science and Engineering, Kyushu Sangyo University, 3‑1 Matsukadai 2‑Chome, Higashi‑ku, Fukuoka 813‑8503, Japan
2
Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 1 M. Kogălniceanu Str., 400084 Cluj‑Napoca, Romania
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in [5] to the case of spiralshapelike domains in ℂn and spirallike domains in ℂn (see [21]). Recent applications of [3, Theorem 2.1] to univalence in higher dimensions may be found in [6, 12, 22, 23, 35]. Arosio, Bracci and Wold (see [6]) used smooth boundary regularity assumptions, to prove that if D is a bounded strictly pseudoconvex domain in ℂn ( n ≥ 2 ) with C∞ boundary, which is biholomorphic to the Euclidean unit ball 𝔹n , then D is convexshapelike (i.e., there exists an automorphism of ℂn which maps D onto a convex domain) if and only if D is polynomially convex. Moreover, using this result, they proved that a normalized biholomorphic mapping of 𝔹n whose image is a smooth strongly pseudoconvex domain with C∞ boundary is embeddable into a normalized filtering Loewner chain with range ℂn if and only if the closure of the image is polynomially convex. In this paper, we first observe that the regularity assumption in [6, Proposition 3.6] can be relaxed to Cm boundary, for some m > 2 + 12 , in view of th
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