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Isometries of intrinsic metrics on strictly convex domains. (English) Zbl 0645.32005

From the authors’ introduction: “We introduce an intrinsic metric g on a complex manifold M admitting a positive, bounded strictly plurisubharmonic function \(\tau\) satisfying the Monge-Ampère equation \(\tau^{\alpha {\bar \beta}}\tau_{\alpha}\tau_{{\bar \beta}}=\tau\). It is shown that the holomorphic sectional curvature of the metric g along the tangential direction of each leaf of the Monge-Ampère foliation associated to \(\tau\) is identically -1. This result applies in particular to bounded strictly convex domains in \({\mathbb{C}}^ n\) with smooth boundaries, for which it is shown that foliation-preserving isometries of the intrinsic metric are biholomorphic or anti- biholomorphic. As a corollary, we obtain a recent theorem of J. Bland, T. Duchamp and M. Kalka [Contemp. Math. 49, 19-30 (1986; Zbl 0589.32050)] that for any two bounded strictly convex domains D and \(\tilde D\) with smooth boundaries, any biholomorphism between corresponding Kobayashi balls \(D_ r\) and \(\tilde D_ r\) having the center fixed can be extended to a biholomorphism of D and \(\tilde D\)”.
Reviewer: M.Stoll

MSC:

32Q99 Complex manifolds
32T99 Pseudoconvex domains

Citations:

Zbl 0589.32050
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References:

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