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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 $$B^n$$.

##### MSC:
 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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##### References:
 [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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