Antonella Nannicini

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

SummaryThe purpose of this paper is to prove that every ellipsoidal domain in Cn admits a complete Kähler metric whose Riemannian sectional curvature is bounded from above by a negative constant (Theorem 1). We construct a Kähler 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 Kähler 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 Kähler 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 authors knowledge, is not known at present if this example is biholomorphic to a domain in Cn.

Generalized Geometry of Norden and Para Norden Manifolds

We study complex structures \(\widehat{J}\) on the generalized tangent bundle of a smooth manifold M compatible with the standard symplectic structure. In particular we describe the class of such generalized complex structures defined by a pseudo Riemannian metric g and a g-symmetric operator H such that H2 = μI, \(\mu \in \mathbb{R}\). These structures include the case of complex Norden manifolds for μ = −1 and the case of Para Norden manifolds for μ = 1 (Nannicini, J Geom Phys 99:244–255, 2016; Nannicini, On a class of pseudo calibrated generalized complex structures: from Norden to para Norden through statistical manifolds, preprint, 2016). We describe integrability conditions of \(\widehat{J}\) with respect to a linear connection ∇ and we give examples of geometric structures that naturally give rise to integrable generalized complex structures. We define the concept of generalized \(\bar{\partial }\mbox{ -}operator\) of (M, H, g, ∇), and we describe certain holomorphic sections. We survey several results appearing in a series of author’s previous papers, (Nannicini, J Geom Phys 56:903–916, 2006; Nannicini, J Geom Phys 60:1781–1791, 2010; Nannicini, Differ Geom Appl 31:230–238, 2013; Adv Geom 16(2):165–173, 2016; Nannicini, Adv Geom 16(2):165–173, 2016; Nannicini, J Geom Phys 99:244–255, 2016; Nannicini, On a class of pseudo calibrated generalized complex structures: from Norden to para Norden through statistical manifolds, preprint, 2016), with special attention to recent results on the generalized geometry of Norden and Para Norden manifolds (Nannicini, J Geom Phys 99:244–255, 2016; Nannicini, Balkan J Geom Appl 22:51–69, 2017).