Quantifies nearly spherical subsets in complex ball geometry.
problem Isoperimetric inequality for nearly spherical domains in Bergman ball.
method Proves a quantitative isoperimetric inequality for nearly spherical subsets of Bergman ball.
result First result on isoperimetric phenomenon in Bergman ball.
The paper characterizes unit balls among Stein spaces with specific groups using Bergman-Einstein metrics.
problem Characterizing unit balls among Stein spaces with specific groups.
method Study of Bergman metric on finite ball quotients and its Kähler-Einstein property.
result The Bergman-Einstein metric exists only on the unit ball itself for finite ball quotients with trivial groups.
Study on when Bergman metrics of domains are induced by balls.
problem When does a domain's Bergman metric match a ball's up to a constant factor?
method Holomorphic isometric immersions, Calabi's diastasis criterion, explicit Bergman kernel formulas, algebraic arguments.
result For strictly pseudoconvex domains in \(\mathbb{C}^2\), if the immersion extends smoothly and transversally past the boundary and the scaling factor meets certain conditions, the domain is biholomorphic to the ball.
Two-dimensional domains with Kähler-Einstein Bergman metrics are biholomorphic to the unit ball.
problem Characterizing domains with Kähler-Einstein Bergman metrics.
method Asymptotics of derivatives of the Bergman kernel along critically tangent paths.
result Two-dimensional pseudoconvex domains with Kähler-Einstein Bergman metrics are biholomorphic to the unit ball.
The paper connects Bergman-Calabi diastasis to Kähler metrics with constant holomorphic sectional curvature.
problem Characterizing domains with Bergman metrics of constant holomorphic sectional curvature.
method Using the Bergman-Calabi diastasis and its connection to the Bergman representative coordinate, the paper derives explicit formulas and proves properties of these metrics.
result Domains with Bergman metrics of constant holomorphic sectional curvature are hyperconvex or exhaustively biholomorphic to a ball.
Study Bergman kernels on complex hyperbolic cusps, generalizing previous results.
problem Localization of Bergman kernels on Kähler manifolds with complex hyperbolic cusps.
method Revisiting Tian's peak section method, applying to Kähler-Einstein metrics and quotients of complex balls.
result Partial localization result for Poincaré type cusps.
Extends Carathéodory's theorem to multidimensional domains with constant curvature.
problem Characterizing biholomorphic domains with constant holomorphic curvature.
method Using Bergman representative coordinates and Calabi's diastasis.
result Provides sufficient conditions for the boundary of a biholomorphic ball to be a topological sphere.
The study shows algebraic Bergman kernels imply finite type boundaries in complex domains.
problem Understanding the relationship between algebraic Bergman kernels and the finite type of boundaries in complex domains.
method Analyzing algebraic Bergman kernels and their implications on the finite type of boundaries in smoothly bounded pseudoconvex domains in C2. result The boundary of a smoothly bounded pseudoconvex domain with an algebraic Bergman kernel of degree d is of finite type with type r≤2d. Local rigidity results for Bergman and Kähler Carathéodory metrics on domains.
problem Characterizing domains with specific metric properties.
method Analyzing Carathéodory and Bergman metrics on strictly pseudoconvex domains.
result Domains with specific metric properties are biholomorphically equivalent to balls.
In this paper we get an explicit lower bound for the radius of a Bergman ball contained in the Dirichlet fundamental polyhedron of a torsion-free discrete group G⊂PU(n,1) acting on complex hyperbolic space. Consequently the volume of all complex hyperbolic n-manifolds is bounded below by the volume of this bal…
Study Bergman metric on Cartan-Hartogs domains and their duals.
problem Characterize Bergman metrics and duals on Cartan-Hartogs domains.
method Define Bergman dual, prove equivalence of conditions, compare with other metrics.
result Equivalence of Bergman metric properties and duals on Cartan-Hartogs domains.
Study on holomorphic isometries between complex domains, revealing geometric properties.
problem Characterizing holomorphic isometries between bounded symmetric domains.
method Analyzing holomorphic isometries between complex unit ball and other bounded symmetric domains, using classical results for complex-analytic subvarieties of Stein manifolds.
result Images of holomorphic isometries have specific geometric properties, including intersections with affine-linear subspaces.
The paper studies the curvature behavior near the boundary of certain domains.
problem Investigating the asymptotic behavior of bisectional curvature for weighted Bergman metrics.
method Characterizing extremal functions via L2-orthogonal projections and using the squeezing function. result The bisectional curvature at strongly pseudoconvex boundary points asymptotically matches that of the unit ball.
The paper proves new inequalities on the unit ball in higher dimensions.
problem Establishing new weighted inequalities on the unit ball.
method Limiting approach to prove Carleman and Huber inequalities.
result Sharp weighted Carleman and Huber inequalities on the unit ball.
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 Cn,n≥2, is Kähler-Einstein if and only if the domain is biholomorphic to the ball. We establish versions of various …
The paper proves a conjecture about the Bergman metric of real analytic domains.
problem Proving the Cheng-Yau conjecture for real analytic pseudoconvex domains.
method Localization of Bergman kernels, extension theorem, and Einstein metrics.
result The Bergman metric of a bounded pseudoconvex domain with real-analytic boundary is Einstein if and only if the domain is biholomorphic to the unit ball.
The paper proves conditions for Kähler-Einstein metrics to remain Kähler-Einstein under cscK perturbations.
problem Conditions for Kähler-Einstein metrics to remain Kähler-Einstein under cscK perturbations.
method Study of constant scalar curvature Kähler (cscK) metrics on complete non-compact Kähler--Einstein manifolds.
result Sufficient conditions for a cscK perturbation of a Kähler--Einstein metric to remain Kähler--Einstein.
We show that the classical Szasz analytic function SN(f)(x) is obtained by applying the pseudo-differential operator f(N−1Dθ) to the Bergman kernels for the Bargmann-Fock space. The expression generalizes immediately to any smooth polarized noncompact complete toric \kahler manifold, defining the generalized S…
In this paper we prove: if a bounded domain with C2 boundary covers a manifold which has finite volume with respect to either the Bergman volume, the Kähler-Einstein volume, or the Kobayashi-Eisenman volume, then the domain is biholomorphic to the unit ball. This answers an old question of Yau. Further, when the dom…
We use the Suita conjecture (now a theorem) to prove that for any domain Ω⊂C its Bergman kernel K(⋅,⋅) satisfies K(z0,z0)=Volume(Ω)−1 for some z0∈Ω if and only if Ω is either a disk minus a (possibly empty) closed polar set or C minus a (possibly empty) …
The paper explores conditions for real holomorphic gradient fields on Kähler and conformally Kähler manifolds.
problem Conditions for real holomorphic gradient fields on Kähler and conformally Kähler manifolds.
method Investigation of real-valued weight functions with real holomorphic gradient fields on Kähler and conformally Kähler manifolds.
result Identification and determination of weight functions with real holomorphic gradient fields on specific metrics.
In this short note we are concerned with the Kahler-Einstein metrics near cone type log canonical singularities. By two different approaches, we construct a complete Kahler-Einstein metric with negative scalar curvature in a neighborhood of the cone over a Calabi-Yau manifold, which provides a local model for the futur…
We prove that every Einstein metric on the unit ball B^4 of C^2, asymptotic to the Bergman metric, is equal to it up to a diffeomorphism. We need a solution of Seiberg--Witten equations in this infinite volume setting. Therefore, and more generally, if M^4 is a manifold with a CR-boundary at infinity, an adapted spinc-…
Formula for Bergman kernel of complex hyperbolic manifolds proved.
problem Calculating the Bergman kernel of complex hyperbolic manifolds.
method Expressed as a sum over geodesic loops.
result Maximum and minimum of the Bergman kernel function proved.
We prove that the quasi-homogenous symbols on the projective space Pn(C) yield commutative algebras of Toeplitz operators on all weighted Bergman spaces, thus extending to this compact case known results for the unit ball Bn. These algebras are Banach but not C∗. We prove the existen…
Study continuity of Bergman kernels on degenerating varieties.
problem Continuity and uniform convergence of Bergman kernels on degenerating varieties.
method Introduced fiberwise Bergman kernel for flat families of polarized varieties, established continuity and uniform convergence results.
result Uniform convergence of Fubini-Study currents and continuity of fiberwise Bergman kernel on test configurations.
Study Bergman kernels on Kähler manifolds, answering Lu-Tian's question.
problem Understanding Bergman kernels on Kähler manifolds and their properties.
method Localization and expansion analysis of Bergman kernels.
result Answered Lu-Tian's question about Bergman kernels having no logarithmic singularity.
The paper proves properties of complex Finsler metrics on specific domains.
problem Investigating invariant complex Finsler metrics on complex domains.
method Analyzing holomorphic automorphism groups and constructing metrics.
result Explicitly constructed metrics on polydisks with properties similar to Bergman metric.
Explicit formula for Bergman kernel of abelian varieties proved.
problem Explicit formula for Bergman kernel of polarized abelian varieties.
method Explicit formula for Bergman kernel of polarized abelian varieties.
result Explicit formula for Bergman kernel of polarized abelian varieties.
The paper studies invariant weighted Bergman metrics on domains.
problem Investigating invariant weighted Bergman metrics under biholomorphisms.
method Introducing invariant weight assignments, using Bergman's minimum integral method and domain version of Tian-Yau-Zelditch expansion.
result Uniform convergence of weighted Bergman kernels and metrics on uniform squeezing domains.
New Schwarz Lemma for Bergman metrics in bounded domains.
problem Finding bounds for Bergman metrics in bounded domains.
method Using Cauchy-Schwarz inequality from probability theory.
result Established a new Schwarz Lemma for Bergman metrics.
Proper holomorphic isometries between Bergman domains are biholomorphisms.
problem Characterizing isometries between Bergman domains.
method New method from Information Geometry.
result Proper holomorphic local isometries are biholomorphisms.
The Bergman metric on symmetrized bidisc has negative curvature properties.
problem Holomorphic curvature properties of the Bergman metric on symmetrized bidisc.
method Analysis of holomorphic sectional and bisectional curvatures.
result Negative pinched holomorphic sectional curvature and non-positive holomorphic bisectional curvature.
Paper compares Bergman kernel and Masur-Veech measure on Teichmüller space.
problem Comparing Bergman kernel and Masur-Veech measure on Teichmüller space.
method Comparison between Bergman kernel form and pushforward measure of Masur-Veech measure.
result Obtained a comparison between the Bergman kernel form and the pushforward measure of the Masur-Veech measure.
The Bergman measure converges to the Zhang measure on a hybrid space.
problem Proving convergence of Bergman measures to Zhang measure.
method Analyzing convergence on a hybrid space and metrized curve complex.
result Bergman measure converges to Zhang measure on a hybrid space.
Study on convergence rate of Bergman metrics on Kähler manifolds.
problem Analyzing convergence rate of Bergman metrics on Kähler manifolds.
method Using Tian's peak section method to show uniform C1,α convergence. result Uniform C1,α convergence of Bergman metrics is demonstrated. We study the asymptotic behavior of the generalized Bergman kernel of the renormalized Bochner-Laplacian on high tensor powers of a positive line bundle on a symplectic manifold of bounded geometry. First, we establish the off-diagonal exponential estimate for the generalized Bergman kernel. As an application, we obtai…
Infinite volumes of Bergman spaces on product manifolds.
problem Volume calculation of Bergman spaces on product manifolds.
method Calabi and Mabuchi volumes, inspired by Shiffman-Zelditch conjecture.
result Infinite volumes of Bergman spaces on product manifolds.
In this paper, we survey some recent results about the asymptotic expansion of Bergman kernel and we give a Bergman kernel proof of Kodaira embedding theorem.
Study shows Bergman metric is non-Einstein for certain domains.
problem Characterizing the Bergman metric of specific domains.
method Analyzing pseudoconvex domains with strongly pseudoconvex polyhedral boundaries.
result Bergman metric is not Einstein for the studied domains.
Unified proofs of weak holomorphic Morse inequalities using Bergman kernel functions.
problem Weak holomorphic Morse inequalities on various types of manifolds.
method Asymptotic Bergman kernel functions and Bochner-Kodaira-Nakano formulas.
result Unified proofs of weak holomorphic Morse inequalities.
New method for spectral and Bergman kernels under local spectral gap condition.
problem Analyzing spectral and Bergman kernels for complex manifolds.
method Developed a new scaling method to study spectral and Bergman kernels.
result Established pointwise asymptotics of spectral and Bergman kernels.
Study Bergman kernel metrics on degenerating hyperelliptic surfaces.
problem Asymptotic behavior of Bergman kernels near singularities.
method Taylor expansion for Abelian differentials and period matrices.
result Explicit coefficients in asymptotic formulas for Bergman kernels.
Develops a universal Hermitian projective calculus for complex hyperbolic two-space
problem Complex hyperbolic geometry
method Algebraic invariant calculus
result Denominator-cleared identities for various geometric quantities
Estimates Bergman kernel for positive line bundles.
problem Estimating the Bergman kernel for positive line bundles.
method Explicit estimate using positive line bundles.
result Explicit lower bound of the Bergman kernel.
Inspired by Berndtsson's work on the subharmonicity property of the Bergman kernel, we give a local variation formula of the full Bergman kernels associated to deformations of complex manifolds. In compact case, it follows from the reproducing property of the Bergman kernel and the curvature formula of the 0-th direct …
Study equivalence of metrics on noncompact Kähler manifolds with Bergman kernel properties.
problem Equivalence of invariant metrics on noncompact Kähler manifolds with bounded curvature.
method Use boundedness of ratio between Bergman kernel and wedge product of metric in fundamental domain.
result Equivalence of Bergman metric and Kähler-Einstein metric under specific conditions.
Study shows Bergman kernel quotient approaches one for punctured surfaces.
problem Analyzing Bergman kernels on punctured Riemann surfaces.
method Examined a punctured Riemann surface with a specific metric and line bundle, calculating quotient of Bergman kernels.
result The quotient of Bergman kernels tends to one as tensor power increases.