Characterizations of the unit ball \(B^n\) in complex Euclidean space. (English) Zbl 0547.32013

Two characterizations are given. It is shown that if \(M\) is a complex manifold such that at one point of \(M\) the Eisenman-Kobayashi and Eisenman- Carathéodory volume forms coincide and are nonzero, then \(M\) is biholomorphic tothe unit ball \(B^n\). This generalizes previous results of B. Wong, Rosay, and Dektyarev. It is also shown, using results of Lempert and Royden - P.M. Wong, that if \(\Omega\) is a strictly convex domain in \({\mathbb{C}}^ n\) such that either (a) there exists a point \(p\in\Omega\) and a holomorphic map \(\tau: B^n\to\Omega\), \(\tau(0)=p,\) such that \(d\tau(0)\) carries the unit ball onto the indicatrix of the Kobayashi metric; or (b) there exists a point \(p\in\Omega \) and a holomorphic map \(\sigma:\Omega\to B^n\), \(\sigma(p)=0\), such that \(d\sigma(p)\) carries the indicatrix of the Carathéodory metric onto the unit ball, then \(\Omega\) is biholomorphic to \(B^n\).


32F45 Invariant metrics and pseudodistances in several complex variables
32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010)
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[1] Barth, T.J.: Convex domains and Kobayashi hyperbolicity. Proc. Am. Math. Soc.79, 556-558 (1980) · Zbl 0438.32013
[2] Bochner, S., Martin, W.T.: Several complex variables, Princeton, N.J.: Princeton Univ. Press 1948 · Zbl 0041.05205
[3] Dektyarev, I.M.: Tests for equivalence of hyperbolic manifolds. Funkts. Anal. i Prilozh.15, 73-74 (1981), (English translation: Funct. Anal. Appl.15, 292-293 (1981) (1982) · Zbl 0478.14017
[4] Eisenman, D.A.: Intrinsic measures on complex manifolds and holomorphic mappings. Mem. Am. Math. Soc., No. 96, Am. Math. Soc., Providence 1970 · Zbl 0197.05901
[5] Graham, I., Wu, H.: Some remarks on the intrinsic measures of Eisenman. To appear · Zbl 0582.32034
[6] Hahn, K.T.: Inequality between the Bergman metric and the Carathéodory differential metric. Proc. Am. Math. Soc.68, 193-194 (1978) · Zbl 0376.32020
[7] Lempert, L.: La métrique de Kobayashi et la representation des domaines sur la boule. Bull. Soc. Math. Fr.109, 427-474 (1981) · Zbl 0492.32025
[8] Look, K.H.: Schwarz lemma and analytic invariants. Sci. Sin.7, 435-504 (1958) · Zbl 0085.06803
[9] Patrizio, G.: Parabolic exhaustions for strictly convex domains, Thesis, University of Notre Dame, 1983 · Zbl 0581.32018
[10] Rosay, J.-P.: Sur une caractérization de la boule parmi les domaines de ? n par son groupe d’automorphismes. Ann. Inst. Fourier29, 91-97 (1979) · Zbl 0402.32001
[11] Royden, H.L., Wong, P.M.: Carathéodory and Kobayashi metric on convex domains. To appear
[12] Stanton, C.: A characterization of the ball by its intrinsic metrics. Math. Ann.264, 271-275 (1983) · Zbl 0511.32001
[13] Wong, B.: Characterization of the unit ball in ? n by its automorphism group. Invent. Math.41, 253-257 (1977) · Zbl 0385.32016
[14] Wu, H.: Normal families of holomorphic mappings. Acta Math.119, 193-233 (1967) · Zbl 0158.33301
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