arXiv research

Study Bergman metric on Cartan-Hartogs domains and their duals.

The Bergman metric on symmetrized bidisc has negative curvature properties.

New Schwarz Lemma for Bergman metrics in bounded domains.

The paper studies invariant weighted Bergman metrics on domains.

Study equivalence of metrics on noncompact Kähler manifolds with Bergman kernel properties.

The paper connects Bergman-Calabi diastasis to Kähler metrics with constant holomorphic sectional curvature.

Study on convergence rate of Bergman metrics on Kähler manifolds.

Study shows Bergman metric is non-Einstein for certain domains.

Local rigidity results for Bergman and Kähler Carathéodory metrics on domains.

Proper holomorphic isometries between Bergman domains are biholomorphisms.

Study shows Bergman kernel quotient approaches one for punctured surfaces.

Two-dimensional domains with Kähler-Einstein Bergman metrics are biholomorphic to the unit ball.

In this paper we will discuss local coordinates canonically corresponding to a Kahler metric. We will also discuss and prove the $C^\infty$ convergence of Bergman metrics following Tian's result on $C^2$ convergence of Bergman metrics. At the end, we present an interesting characterization of ample line bundle that cou…

Study on when Bergman metrics of domains are induced by balls.

The paper characterizes unit balls among Stein spaces with specific groups using Bergman-Einstein metrics.

In this paper we consider a punctured Riemann surface endowed with a Hermitian metric which equals the Poincaré metric near the punctures and a holomorphic line bundle which polarizes the metric. We show that the Bergman kernel can be localized around the singularities and its local model is the Bergman kernel of the p…

Study Bergman kernel metrics on degenerating hyperelliptic surfaces.

Study on symmetric domains with Bergman metric properties.

We prove that the Bergman and the Teichmuller metrics are equivalent on Teichmuller spaces.

Study shows quantum behavior near infinity in metric asymptotics.

Rigidity theorem for Bergman metric on Hartogs domains over bounded homogeneous domains.

Study Bergman kernels on complex hyperbolic cusps, generalizing previous results.

The paper connects Bergman geometry with information geometry.

New metric defined for bounded symmetric domains.

Consider the sum of the first $N$ eigenspaces for the Laplacian on a Riemannian manifold. A basis for this space determines a map to Euclidean space and for $N$ sufficiently large the map is an embedding. In analogy with a fruitful idea of Kähler geometry, we define (Riemannian) Bergman metrics of degree $N$ to be thos…

Let $X$ be a compact hyperbolic Riemann surface equipped with the Poincaré metric. For any integer $k\geq 2$, we investigate the Bergman kernel associated to the holomorphic Hermitian line bundle $Ω^{\otimes k}_X$, where $Ø$ is the holomorphic cotangent bundle of $X$. Our first main result estimates the corresponding B…

The paper introduces new metrics on complex domains with specific geometric properties.

Using the techniques developed in \cite{SunSun}, we give estimations of the Bergman kernel of the punctured disk with the standard complete Poincaré metric. As an application, we improve the result of \cite{AMM} on the Bergman kernels of punctured Riemann surfaces near singularities.

The paper proves a conjecture about the Bergman metric of real analytic domains.

We show that the exponential map of the Bochner connection on the restricted holomorphic tangent bundle of a complex manifold admitting the positive-definite Bergman metric coincides with the inverse of Bergman's representative map. We also present a generalization of the Lu theorem, as an application.

Study Bergman kernels on Kähler manifolds, answering Lu-Tian's question.

We show that under very general assumptions the partial Bergman kernel function of sections vanishing along an analytic hypersurface has exponential decay in a neighborhood of the vanishing locus. Considering an ample line bundle, we obtain a uniform estimate of the Bergman kernel function associated to a singular metr…

We study partition functions of random Bergman metrics, with the actions defined by a class of geometric functionals known as `stability functions'. We introduce a new stability invariant - the critical value of the coupling constant - defined as the minimal coupling constant for which the partition function converges.…

The paper studies the curvature behavior near the boundary of certain domains.

Sharp estimates for Bergman metrics derived from Kähler quantization.

This article contains a detailed study, in the toric case, of the test configuration geodesic rays defined by Phong-Sturm. We show that the `Bergman approximations' of Phong-Sturm converge in C^1 to the geodesic ray and that the geodesic ray itself is C^{1,1} and no better. The \kahler metrics associated to the geodesi…

Geodesics on the infinite dimensional symmetric space $\hcal$ of Kähler metrics in a fixed Kähler class on a projective Kähler manifold X are solutions of a homogeneous complex Monge-Ampère equation in $X \times A$, where $A \subset \C$ is an annulus. They are analogues of 1PS (one-parameter subgroups) on symmetric spa…

Without using the $L^2$ extension theorem, we provide a new proof of the equality part in Suita's conjecture, which states that for any open Riemann surface admitting a Green's function, the Bergman kernel and the logarithmic capacity coincide at one point if and only if the surface is biholomorphic to a disc possibly …

In this paper, we study the behavior of Bergman kernels along the Kähler Ricci flow on Fano manifolds. We show that the Bergman kernels are equivalent along the Kähler Ricci flow under certain condition on the Ricci curvature of the initial metric. Then, using a recent work of Tian and Zhang, we can solve a conjecture …

The space of positively curved hermitian metrics on a positive holomorphic line bundle over a compact complex manifold is an infinite-dimensional symmetric space. It is shown by Phong and Sturm that geodesics in this space can be uniformly approximated by geodesics in the finite dimensional spaces of Bergman metrics. W…

Extends Carathéodory's theorem to multidimensional domains with constant curvature.

We give an alternate proof of the existence of the asymptotic expansion of the Bergman kernel associated to the $k$-th tensor powers of a positive line bundle $L$ in a $\frac{1}{\sqrt{k}}$-neighborhood of the diagonal using elementary methods. We use the observation that after rescaling the Kähler potential $k\varphi$ …

We give a purely complex geometric proof of the existence of the Bergman kernel expansion. Our method provides a sharper estimate, and in the case that the metrics are real analytic, we prove that the remainder decays faster than any polynomial.

It's well-known in \kahler geometry that the infinite dimensional symmetric space $\hcal$ of smooth \kahler metrics in a fixed \kahler class on a polarized \kahler manifold is well approximated by finite dimensional submanifolds $\bcal_k \subset \hcal$ of Bergman metrics of height $k$. Then it's natural to ask whether …

New geometric conditions ensure compactness of $\bar{\partial}$-Neumann problem.

Study Bergman kernels on Kähler orbifolds with specific properties.

Smooth families of biholomorphisms between strongly pseudoconvex domains are shown to be smooth.

The paper proves the existence of a special type of metric on complex manifolds.