On the existence of complete Kähler metrics of negative Riemannian curvature bounded away from zero on ellipsoidal domains inC n

Abstract

The purpose of this paper is to prove that every ellipsoidal domain in Cn admits a complete Kahler metric whose Riemannian sectional curvature is bounded from above by a negative constant (Theorem 1). We construct a Kahler metric, in a natural way, as potential of a suitable function defining the boundary (§2). Directly we compute the curvature tensor and we find upper and lower bounds for the holomorphic sectional curvature (§ 4, § 5). In order to prove the boundness of Riemannian sectional curvature we use finally a classical pinching argument (§ 6). We also obtain that for certain ellipsoidal domains the curvature tensor is very strongly negative in the sense of [15] (§ 3). Finally we prove that the metric constructed on ellipsoidal domains in Cn is the Bergman metric if and only if the domain is biholomorphic to the ball (Theorem 2). In [8], [9] R. E. Greene and S. G. Krantz gave large families of examples of complete Kahler manifolds with Riemannian sectional curvature bounded from above by a negative constant; they are sufficiently small deformations of the ball in Cn, with the Bergman metric. Before the only known example of complete simply-connected Kahler manifold with Riemannian sectional curvature upper bounded by a negative constant, not biholomorphic to the ball, was the surface constructed by G. D. Mostow and Y. T. Siu in [14], to the best of the author's knowledge, is not known at present if this example is biholomorphic to a domain in Cn.