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A simple proof of a theorem of Calabi. (English) Zbl 1279.32009

Let \(S\) denote one of the complex space forms \(\mathbb C^n\), \(\mathbb P^n\), \(\mathbb B^n\), each equipped with its standard metric (Euclidean, Fubini-Study, Poincaré). Let \(M\) be a connected complex manifold with real analytic Kähler metric, \(U\subset M\) a connected open subset and \(F:U\rightarrow S\) a holomorphic isometric embedding. The authors give a short and more or less straightforward proof (carried out for \(S=\mathbb P^n\)) that \(F\) can be holomorphically extended along every continuous curve \(\gamma: [0,1]\rightarrow M\) with \(\gamma(0)\in U\); in particular \(F\) admits a holomorphic extension \(M\rightarrow S\) if \(M\) is simply connected. Moreover, any isometric holomorphic embedding \(G:U\rightarrow S\) has the form \(G=T\circ F\) with some holomorphic isometry \(T\) of \(S\). The original proof for these facts, also valid for a Hilbert space form \(S\), was given by E. Calabi [Ann. Math. (2) 58, 1-23 (1953; Zbl 0051.13103)]. For recent work in this connection see also N. Mok and S.-C. Ng [Reine Angew. Math. 669, 47–73 (2012; Zbl 1254.32033)].

MSC:

32D15 Continuation of analytic objects in several complex variables
53C55 Global differential geometry of Hermitian and Kählerian manifolds
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
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Full Text: Euclid