Let $M\subset{\complex}^n$ be a complex domain of ${\complex}^n$ endowed with a rotation invariant \K form $\omega_{\Phi}= \frac{i}{2} \partial\bar\partial\Phi$. In this paper we describe sufficient conditions on the \K potential $\Phi$ for $(M, \omega_{\Phi})$ to admit a symplectic embedding (explicitely described in terms of $\Phi$) into a complex space form of the same dimension of $M$. In particular we also provide conditions on $\Phi$ for $(M, \omega_{\Phi})$ to admit global symplectic coordinates. As an application of our results we prove that each of the Ricci flat (but not flat) \K forms on ${\complex}^2$ constructed by LeBrun (Taub-NUT metric) admits explicitely computable global symplectic coordinates.Comment: to appear in Journal of Geometry and Physic

In this article, we study the set of balanced metrics given in Donaldson's terminology (J. Diff. Geometry 59:479-522, 2001) on a compact complex manifold M which are homothetic to a given balanced one. This question is related to various properties of the Tian-Yau-Zelditch approximation theorem for Kähler metrics. We prove that this set is finite when M admits a non-positive Kähler-Einstein metric, in the case of non-homogenous toric Kähler-Einstein manifolds of dimension ≤ 4 and in the case of the constant scalar curvature metrics found in Arezzo and Pacard (Acta. Math. 196(2):179-228, 2006; Ann. Math. 170(2):685-738, 2009). © 2011 Springer Science+Business Media B.V

Abstract. Inspired by the work of G. Lu [34] on pseudo symplectic capacities we obtain several results on the Gromov width and the Hofer-Zehnder capacity of Hermitian symmetric spaces of compact type. Our results and proofs extend those obtained by Lu for complex Grassmannians to Hermitian symmetric spaces of compact type. We also compute the Gromov width and the Hofer-Zehnder capacity for Cartan domains and their products.

We propose two conjectures about Ricci-flat Kähler metrics:Conjecture 1: A Ricci-flat projectively induced metric is flat.Conjecture 2: A Ricci-flat metric on an n-dimensional complex manifold such that the a n+1 coefficient of the TYZ expansion vanishes is flat.We verify Conjecture 1 (see Theorem 1.1) under the assumptions that the metric is radial and stable-projectively induced and Conjecture 2 (see Theorem 1.2) for complex surfaces whose metric is either radial or complete and ALE. We end the paper by showing, by means of the Simanca metric, that the assumption of Ricci-flatness in Conjecture 1 and in Theorem 1.2 cannot be weakened to scalar-flatness (see Theorem 1.3).2010 Mathematics Subject Classification. 53C55; 58C25; 58F06.

We classify radial scalar flat metrics with constant third coeffcient of its TYZ expansion. As a byproduct of our analysis we provide a characterization of Simanca's scalar flat metric.2010 Mathematics Subject Classification. 53C55; 58C25; 58F06.