Related papers: Bergman--Einstein Rigidity for Hartogs Domains ove…

We study the Bergman metric and introduce the Bergman dual on Cartan-Hartogs (CH) domains. For a bounded domain D in C^n with Bergman kernel K_D, we define the Bergman dual of (D, g_D) as (D*, g_D*), where D* is the maximal domain on which…

We study bounded domains $\Omega\subset\mathbb{C}^n$ whose Bergman metric is locally symmetric, i.e. its Riemannian curvature tensor is parallel with respect to the Levi-Civita connection. Following the strategy developed in…

In this paper, we first establish the localization of the Bergman kernels for unbounded pseudoconvex domains near a D'Angelo finite type boundary point. This result was proved by Engli\v{s} more than twenty years ago for bounded…

We use the Suita conjecture (now a theorem) to prove that for any domain $\Omega \subset \mathbb{C}$ its Bergman kernel $K(\cdot, \cdot)$ satisfies $K(z_0, z_0) = \hbox{Volume}(\Omega)^{-1}$ for some $z_0 \in \Omega$ if and only if $\Omega$…

The Fefferman--Szeg\H{o} metric \(g_{\operatorname{FS}}^\Omega\) on a \(C^\infty\)-smooth bounded strongly pseudoconvex domain \(\Omega\subset\mathbb C^n\) is an invariant metric defined via the Fefferman surface measure. For this metric,…

The Cartan-Hartogs domains are defined as a class of Hartogs type domains over irreducible bounded symmetric domains. The purpose of this paper is twofold. Firstly, for a Cartan-Hartogs domain $\Omega^{B^{d_0}}(\mu)$ endowed with the…

In this new version, we give an affirmative solution to a conjecture of Cheng proposed in 1979 which asserts that the Bergman metric of a smoothly bounded strongly pseudoconvex domain in $\mathbb{C}^n, n\geq 2,$ is K\"ahler-Einstein if and…

Let $\Omega$ be a Stein space with a compact smooth strongly pseudoconvex boundary. We prove that the boundary is spherical if its Bergman metric over $\hbox{Reg}(\Omega)$ is K\"ahler-Einstein.

In this paper, we study constant scalar curvature K\"ahler (cscK) metrics on complete non-compact K\"ahler--Einstein manifolds. We give sufficient conditions under which a cscK perturbation of a K\"ahler--Einstein metric must remain…

Let $\Omega$ be a convex domain in $\mathbb{C}^n$ and $\varphi$ a convex function on $\Omega$. We prove that $\log{K_{\Omega,\varphi}(z)}$ is a convex function (might be identically $-\infty$) on $\Omega$, where $K_{\Omega,\varphi}$ is the…

The $p-$Bergman kernel $K_p(\cdot)$ is shown to be of $C^{1,1/2}$ for $1<p<\infty$. An unexpected relation between the off-diagonal $p-$Bergman kernel $K_p(\cdot,z)$ and certain weighted $L^2$ Bergman kernel is given for $1\le p\le 2$. As…

We investigate when the Bergman metric of a bounded domain is, up to a constant factor $\lambda$, induced by the Bergman metric of a finite-dimensional unit ball $\mathbb{B}^N$ via a holomorphic isometric immersion. For a strictly…

We study the complete K\"{a}hler-Einstein metric of a Hartogs domain $\widetilde {\Omega}$, which is obtained by inflation of an irreducible bounded symmetric domain $\Omega $, using a power $N^{\mu}$ of the generic norm of $\Omega$. The…

The Fock-Bargmann-Hartogs domain $D_{n,m}(\mu)$ ($\mu>0$) in $\mathbf{C}^{n+m}$ is defined by the inequality $\|w\|^2<e^{-\mu\|z\|^2},$ where $(z,w)\in \mathbf{C}^n\times \mathbf{C}^m$, which is an unbounded non-hyperbolic domain in…

In this note we shall prove that the complete K\"{a}hler-Einstein volume form on a bounded strongly pseudoconvex domain with $C^{\infty}$-boundary is the normalized limit of a sequence of Bergman kernels.

We study equivalence of invariant metrics on noncompact K\"ahler manifolds with a complete Bergman metric of bounded curvature. Especially only the boundedness of the ratio between Bergman kernel and the $n$-times wedge product of Bergman…

We show that the $p-$Bergman kernel $K_p(z)$ on a bounded domain $\Omega$ is of locally $C^{1,1}$ for $p\geq1$.The proof is based on the locally Lipschitz continuity of the off-diagonal $p-$Bergman kernel $K_p(\zeta,z)$ for fixed $\zeta\in…

We study the density of functions which are holomorphic in a neighbourhood of the closure $\overline{\Omega}$ of a bounded non-smooth pseudoconvex domain $\Omega$, in the Bergman space $ H^2(\Omega ,\varphi)$ with a plurisubharmonic weight…

We study the rigidity problem for $(-\alpha)$-homogeneous solutions to the two-dimensional incompressible stationary Euler equations in sector-type domains $\Omega_{a, b, \theta_0}:= \{(r,\theta): a<r<b, \ 0<\theta<\theta_0\}$, where…

We prove that a two dimensional pseudoconvex domain of finite type with a K\"ahler-Einstein Bergman metric is biholomorphic to the unit ball. This answers an old question of Yau for such domains. The proof relies on asymptotics of…