We study local rigidity and multiplicity of constant scalar curvature metrics in arbitrary products of compact manifolds. Using (equivariant) bifurcation theory we determine the existence of infinitely many metrics that are accumulation points of pairwise non-homothetic solutions of the Yamabe problem. Using local rigidity and some compactness results for solutions of the Yamabe problem, we also exhibit new examples of conformal classes (with positive Yamabe constant) for which uniqueness holds

Abstract. This paper consists of two results dealing with balanced metrics (in S. Donaldson terminology) on nonconpact complex manifolds. In the first one we describe all balanced metrics on Cartan domains. In the second one we show that the only Cartan-Hartogs domain which admits a balanced metric is the complex hyperbolic space. By combining these results with those obtained in [13] we also provide the first example of complete, Kähler-Einstein and projectively induced metric g such that αg is not balanced for all α > 0.

The study of Kähler immersions of a given real analytic Kähler 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 Kähler 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 Kähler immersion into another. Although, a complete classification of Kähler manifolds admitting a Kähler immersion into complex space forms is not known, not even when the Kähler manifolds involved are of great interest, e.g. when they are Kähler-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.Each chapter begins with a brief summary of the topics discussed and ends with a list of exercises which help the reader to test his understanding.Apart from the topics discussed in Section 3.1 of Chapter 3, which could be skipped without compromising the understanding of the rest of the book, the requirements to read this book are a basic knowledge of complex and Kähler iii geometry (treated, e.g. in Moroianu's book [59]).The authors are grateful to Claudio Arezzo and Fabio Zuddas for a careful reading of the text and for valuable comments that have improved the book's exposure.

This paper consists of two main results. In the first one we describe all Kähler immersions of a bounded symmetric domain into the infinite dimensional complex projective space in terms of the Wallach set of the domain. In the second one we exhibit an example of complete and nonhomogeneous Kähler-Einstein metric with negative scalar curvature which admits a Kähler immersion into the infinite dimensional complex projective space.

In this paper we address two problems concerning a family of domains M Ω (µ) ⊂ C n , called Cartan-Hartogs domains, endowed with a natural Kähler metric g(µ). The first one is determining when the metric g(µ) is extremal (in the sense of Calabi), while the second one studies when the coefficient a 2 in the Engliš expansion of Rawnsley ε-function associated to g(µ) is constant.