Stenzel's Ricci-flat Kaehler metrics are not projectively induced

We investigate the existence of a holomorphic and isometric immersion in the complex projective space for the complete Ricci-flat Kaehler metrics constructed by M. B. Stenzel on the cotangent bundle of a compact, rank one, globally symmetric space.Comment: 8 page

On the convergence of the Sasaki J-flow

This paper investigates the $C^\infty$-convergence of the Sasaki $J$-flow. The result is applied to prove a lower bound for the $K$-energy map in the Sasakian context.Comment: 15 page

A note on the coefficients of Rawnsley's epsilon function of Cartan-Hartogs domains

We extend a result of Z. Feng and Z. Tu by showing that if one of the coefficients $a_j$, $2\leq j\leq n$, of Rawnlsey's epsilon function associated to a $n$-dimensional Cartan-Hartogs domain is constant, then the domain is biholomorphically equivalent to the complex hyperbolic space.Comment: 6 p

Canonical metrics on Cartan--Hartogs domains

In this paper we address two problems concerning a family of domains M_{\Omega}(\mu) \subset \C^n, called Cartan-Hartogs domains, endowed with a natural Kaehler metric $g(\mu)$. The first one is determining when the metric $g(\mu)$ is extremal (in the sense of Calabi), while the second one studies when the coefficient $a_2$ in the Engli\v{s} expansion of Rawnsley $\epsilon$-function associated to $g(\mu)$ is constant.Comment: 13 page

K\"ahler immersions of K\"ahler manifolds into complex space forms

The study of K\"ahler immersions of a given real analytic K\"ahler manifold into a finite or infinite dimensional complex space form originates from the pioneering work of Eugenio Calabi [10]. With a stroke of genius Calabi defines a powerful tool, a special (local) potential called diastasis function, which allows him to obtain necessary and sufficient conditions for a neighbourhood of a point to be locally K\"ahler immersed into a finite or infinite dimensional complex space form. As application of its criterion, he also provides a classification of (finite dimensional) complex space forms admitting a K\"ahler immersion into another. Although, a complete classification of K\"ahler manifolds admitting a K\"ahler immersion into complex space forms is not known, not even when the K\"ahler manifolds involved are of great interest, e.g. when they are K\"ahler-Einstein or homogeneous spaces. In fact, the diastasis function is not always explicitely given and Calabi's criterion, although theoretically impeccable, most of the time is of difficult application. Nevertheless, throughout the last 60 years many mathematicians have worked on the subject and many interesting results have been obtained. The aim of this book is to describe Calabi's original work, to provide a detailed account of what is known today on the subject and to point out some open problems.Comment: 116 page

Isometric immersions of locally conformally Kaehler manifolds

We investigate isometric immersions of locally conformally Kaehler metrics into Hopf manifolds. In particular, we study Hopf-induced metrics on compact complex surfaces.Comment: 20 pages. Revised version with main corrections to Prop. 3.4, Prop. 3.6 and Prop. 3.9 (and Th. 1.1 consequently

Stenzel's Ricci-flat Kähler metrics are not projectively induced

We investigate the existence of a holomorphic and isometric immersion in the complex projective space for the complete Ricci-flat Kaehler metrics constructed by M. B. Stenzel on the cotangent bundle of a compact, rank one, globally symmetric space

A Note on the L2-norm of the Second Fundamental Form of Algebraic Manifolds

2000 Mathematics Subject Classification: 53C42, 53C55.Let M f → CP^n be an algebraic manifold of complex dimension d and let σf be its second fundamental form. In this paper we address the following conjecture, which is the analogue of the one stated by M. Gromov for smooth immersions: ... We prove the conjecture in the following three cases: (i) d = 1; (ii) M is a complete intersection; (iii) the scalar curvature of M is constant