In this paper we study the complete invariant metrics on CartanHartogs domains which are the special types of Hua domains. Firstly, we introduce a class of new complete invariant metrics on these domains, and prove that these metrics are equivalent to the Bergman metric. Secondly, the Ricci curvatures under these new metrics are bounded from above and below by the negative constants. Thirdly, w...

In this primarily expository paper we study the analysis of the Diederich-Fornæss worm domain in complex Euclidean space. We review its importance as a domain with nontrivial Nebenhülle, and as a counterexample to a number of basic questions in complex geometric analysis. Then we discuss its more recent significance in the theory of partial differential equations: the worm is the first smoothly...

The Ricci curvature of the Bergman metric on a bounded domain $D\subset \mathbb{C}^n$ is strictly above by $n+1$ and consequently $\log (K_D^{n+1}g_{B,D})$, where $K_D$ kernel for $D$ diagonal $g_{B, D}$ Riemannian volume element $D$, potential K\"ahler known as Kobayashi--Fuks metric. In this note we study localization near holomorphic peak points also show that shares several properties with ...

Consider the zero set of a random power series ∑ anz n with i.i.d. complex Gaussian coefficients an. We show that these zeros form a determinantal process: more precisely, their joint intensity can be written as a minor of the Bergman kernel. We show that the number of zeros in a disk of radius r about the origin has the same distribution as the sum of independent {0, 1}-valued random variables...

We establish asymptotic formulas for polynomials that are orthogonal over the unit disk with respect to a weight of the form |w(z)|2, where w(z) is a polynomial without zeros on the unit circle |z| = 1. The formulas put in evidence a strong connection between the behavior of the polynomials and the reproducing kernel of an associated weighted Bergman space, which produces interesting new featur...

Let D be a pseudoconvex domain in Ckt × C n z and let φ be a plurisubharmonic function in D. For each t we consider the ndimensional slice of D, Dt = {z; (t, z) ∈ D}, let φ be the restriction of φ to Dt and denote by Kt(z, ζ) the Bergman kernel of Dt with the weight function φ. Generalizing a recent result of Maitani and Yamaguchi (corresponding to n = 1 and φ = 0) we prove that logKt(z, z) is ...

Let Sp(n,R) be the symplectic group, and K∗ n its maximal compact subgroup. Then G = Sp(n,R)/K∗ n can be realized as the Siegel domain of type one. The square-integrable representation of G gives the admissible wavelets AW and wavelet transform. The characterization of admissibility condition in terms of the Fourier transform is given. The Bergman kernel follows from the viewpoint of coherent s...

We study the near diagonal asymptotic expansion of the generalized Bergman kernel of the renormalized Bochner-Laplacian on high tensor powers of a positive line bundle over a compact symplectic manifold. We show how to compute the coefficients of the expansion by recurrence and give a closed formula for the first two of them. As consequence, we calculate the density of states function of the Bo...

Let f : X −→ S be a smooth projective family and let (L, h) be a singular hermitian line bundle on X with semipositive curvature current. Let Ks := K(Xs, KXs +L | Xs, h | Xs)(s ∈ S) be the Bergman kernel of KXs +L | Xs with respect to h | Xs and let hB the singular hermitian metric on KX + L defined by hB |Xs := 1/Ks. We prove that hB has semipositive curvature. This is a generalization of the ...