Bergman Metric Research Articles

Metrics on complex manifolds

In this note we discuss various canonical metrics on complex manifolds. A generalization of the Bergman metric is proposed and the relations of metrics on moduli spaces are commented. At last, we review some existence theorems of solutions to the Strominger system.

On holomorphic isometric embeddings of the unit n-ball into products of two unit m-balls

We study holomorphic isometric embeddings of the complex unit n-ball into products of two complex unit m-balls with respect to their Bergman metrics up to normalization constants (the isometric constant). There are two trivial holomorphic isometric embeddings for m ≥ n, given by F1(z) = (0, In;m(z)) with the isometric constant equal to (m + 1)/(n + 1) and F2(z) = (In;m(z), In;m(z)) with the isometric constant equal to 2(m + 1)/(n + 1). Here \({I_{n;m}:\mathbb{C}^n \longrightarrow \mathbb{C}^m}\) is the canonical embedding. We prove that when m < 2n, these are the only holomorphic isometric embeddings up to unitary transformations.

BERGMAN KERNELS AND EQUILIBRIUM MEASURES FOR POLARIZED PSEUDO-CONCAVE DOMAINS

Let X be a domain in a closed polarized complex manifold (Y,L), where L is a (semi-) positive line bundle over Y. Any given Hermitian metric on L induces by restriction to X a Hilbert space structure on the space of global holomorphic sections on Y with values in the k-th tensor power of L (also using a volume form ωn on X. In this paper the leading large k asymptotics for the corresponding Bergman kernels and metrics are obtained in the case when X is a pseudo-concave domain with smooth boundary (under a certain compatibility assumption). The asymptotics are expressed in terms of the curvature of L and the boundary of X. The convergence of the Bergman metrics is obtained in a more general setting where (X,ωn) is replaced by any measure satisfying a Bernstein–Markov property. As an application the (generalized) equilibrium measure of the polarized pseudo-concave domain X is computed explicitly. Applications to the zero and mass distribution of random holomorphic sections and the eigenvalue distribution of Toeplitz operators will be described elsewhere.

Bergman approximations of harmonic maps into the space of Kahler metrics on toric varieties

We generalize the results of Song–Zelditch on geodesics in spaces of Kahler metrics on toric varieties to harmonic maps of any compact Riemannian manifold with boundary into the space of Kahler metrics on a toric variety. We show that the harmonic map equation can always be solved and that such maps may be approximated in the $C2$ topology by harmonic maps into the spaces of Bergman metrics. In particular, Wess–Zumino–Witten (WZW) maps, or equivalently solutions of a homogeneous Monge–Ampere equation on the product of the manifold with a Riemann surface with $S1$ boundary admit such approximations. We also show that the Eells–Sampson flow on the space of Kahler potentials is transformed to the usual heat flow on the space of symplectic potentials under the Legendre transform, and hence it exists for all time and converges.

An example for the holomorphic sectional curvature of the Bergman metric

We study the behaviour of the holomorphic sectional curvature (or Gaussian curvature) of the Bergman metric of planar annuli. The results are then utilized to construct a domain for which the curvature is divergent at one of its boundary points and moreov

Second fundamental forms of holomorphic isometries of the Poincaré disk into bounded symmetric domains and their boundary behavior along the unit circle

Motivated by problems arising from Arithmetic Geometry, in an earlier article one of the authors studied germs of holomorphic isometries between bounded domains with respect to the Bergman metric. In the case of a germ of holomorphic isometry f: (Δ, λ dsΔ2;0) → (Ω, dsΩ2;0) of the Poincare disk Δ into a bounded symmetric domain Ω ⋐ ℂN in its Harish-Chandra realization and equipped with the Bergman metric, f extends to a proper holomorphic isometric embedding F: (Δ, λ dsΔ2;) → (Ω, dsΩ2) and Graph(f) extends to an affine-algebraic variety V ⊂ ℂ × ℂN. Examples of F which are not totally geodesic have been constructed. They arise primarily from the p-th root map ρp: H → Hp and a non-standard holomorphic embedding G from the upper half-plane to the Siegel upper half-plane H3 of genus 3. In the current article on the one hand we examine second fundamental forms σ of these known examples, by computing explicitly φ = |σ|2. On the other hand we study on the theoretical side asymptotic properties of σ for arbitrary holomorphic isometries of the Poincare disk into polydisks. For such mappings expressing via the inverse Cayley transform in terms of the Euclidean coordinate τ = s + it on the upper half-plane H, we have φ(τ) = t2u(τ), where u|t=0 ≢ 0. We show that u must satisfy the first order differential equation δu/δt|t=0 ≡ 0 on the real axis outside a finite number of points at which u is singular. As a by-product of our method of proof we show that any non-standard holomorphic isometric embedding of the Poincare disk into the polydisk must develop singularities along the boundary circle. The equation δu/δt|t=0 ≡ 0 along the real axis for holomorphic isometries into polydisks distinguishes the latter maps from holomorphic isometries into Siegel upper half-planes arising from G. Towards the end of the article we formulate characterization problems for holomorphic isometries suggested both by the theoretical and the computational results of the article.

On the Equicontinuity Region of Discrete Subgroups of PU(1,n)

Let G be a discrete subgroup of PU(1,n). Then G acts on ℙ nℂ preserving the unit ball ℍ nℂ , where it acts by isometries with respect to the Bergman metric. In this work we look at its action on all of ℙ nℂ and determine its equicontinuity region Eq(G). This turns out to be the complement of the union of all complex projective hyperplanes in ℙ nℂ which are tangent to ∂ℍ nℂ at points in the Chen-Greenberg limit set Λ(G), a closed G-invariant subset of ∂ℍ nℂ which is minimal for non-elementary groups. We also prove that the action on Eq(G) is discontinuous. Also , if the limit set is “sufficiently general” (i.e. it is not contained in any proper k -chain), then each connected component of Eq(G) is a holomorphy domain and it is a complete Kobayashi hyperbolic space.

Composition type operators from Hardy spaces to μ-Bloch spaces on the unit ball

Let φ be a holomorphic self-map of B n and ψ ∈ H( B n ). A composition type operator is defined by T ψ,φ( f)= ψ f o φ for f ∈ H( B n ), which is a generalization of the multiplication operator and the composition operator. In this article, the necessary and sufficient conditions are given for the composition type operator T ψ,φ to be bounded or compact from Hardy space H p ( B n ) to μ-Bloch space B μ( B n ). The conditions are some supremums concerned with ψ, φ, their derivatives and Bergman metric of B n . At the same time, two corollaries are obtained.

Holomorphic invariant forms of a bounded domain

Given a complete ortho-normal system ϕ = (ϕ0,ϕ1,ϕ2,…) of L 2 H(\( \mathcal{D} \)), the space of holomorphic and absolutely square integrable functions in the bounded domain \( \mathcal{D} \) of ℂ n , we construct a holomorphic imbedding \( \iota _\phi :\mathcal{D} \to \mathfrak{F}(n,\infty ) \), the complex infinite dimensional Grassmann manifold of rank n. It is known that in \( \mathfrak{F}(n,\infty ) \) there are n closed (μ, μ)-forms τμ (μ = 1,…,n) which are invariant under the holomorphic isometric automorphism of \( \mathfrak{F}(n,\infty ) \) and generate algebraically all the harmonic differential forms of \( \mathfrak{F}(n,\infty ) \). So we obtain in \( \mathcal{D} \) a set of (μ, μ)-forms ι ϕ * τμ (μ = 1,…, n), which are independent of the system ϕ chosen and are invariant under the bi-holomorphic transformations of \( \mathcal{D} \). Especially the differential metric ds 2 1 associated to the Kahler form ι ϕ * τ1 is a Kahler metric which differs from the Bergman metric ds 2 of \( \mathcal{D} \) in general, but in case that the Bergman metric is an Einstein metric, ds 1 2 differs from ds 2 only by a positive constant factor.

Generalized Berezin quantization, Bergman metrics and fuzzy laplacians

We study extended Berezin and Berezin-Toeplitz quantization for compact K?hler manifolds, two related quantization procedures which provide a general framework for approaching the construction of fuzzy compact K?hler geometries. Using this framework, we show that a particular version of generalized Berezin quantization, which we baptize ``Berezin-Bergman quantization'', reproduces recent proposals for the construction of fuzzy K?hler spaces. We also discuss how fuzzy laplacians can be defined in our general framework and study a few explicit examples. Finally, we use this approach to propose a general explicit definition of fuzzy scalar field theory on compact K?hler manifolds.

Remarks on two theorems of Qi-Keng Lu

Two alternate arguments in the framework of intrinsic metrics and measures respectively of part of the proof of a famous theorem due to Qi-Keng Lu on Bergman metric with constant negative holomorphic sectional curvature are presented. A relationship between the Lu constant and the holomorphic sectional curvature of the Bergman metric is given. Some recent progress of the Yau’s porblem on the characterization of domain of holomorphy on which the Bergman metric is Kahler-Einstein is described.

Derivative-free characterizations of Qk spaces

AbstractWe give two charaterizations of the Möbius invariant QK spaces, one in terms of a double integral and the other in terms of the mean oscillation in the Bergman metric. Both charaterizations avoid the use of derivatives. Our results are new even in the case of Qp.

The equivalence on classical metrics

In this paper we study the complete invariant metrics on Cartan-Hartogs 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, we estimate the holomorphic sectional curvatures of the new metrics, and prove that the holomorphic sectional curvatures are bounded from above and below by the negative constants. Finally, by using these new metrics and Yau's Schwarz lemma we prove that the new metrics are equivalent to the Einstein-Kahler metric. That means that the Yau's conjecture is true on Cartan-Hartogs domains.

A Note on the Bergman metric of Bounded homogeneous Domains

AbstractWe show that the Bergman metric of a bounded homogeneous domain has a potential function whose gradient has a constant norm with respect to the Bergman metric, and further that this constant is independent of the choice of such a potential function.

A new invariant Kähler metric on relatively compact domains in a complex manifold

We introduce a new invariant Kähler metric on relatively compact domains in a complex manifold, which is the Bergman metric of the $L^2$ space of holomorphic sections of the pluricanonical bundle equipped with the Hermitian metric introduced by Narasimhan

An example of a pseudoconvex domain whose holomorphic sectional curvature of the Bergman metric is unbounded

An example of a pseudoconvex domain whose holomorphic sectional curvature of the Bergman metric is unbounded

Test configurations for K-stability and geodesic rays

Let $X$ be a compact complex manifold, $L\to X$ an ample line bundle over $X$, and ${\cal H}$ the space of all positively curved metrics on $L$. We show that a pair $(h_0,T)$ consisting of a point $h_0\in {\cal H}$ and a test configuration $T=({\cal L}\to {\cal X}\to {\bf C})$, canonically determines a weak geodesic ray $R(h_0,T)$ in ${\cal H}$ which emanates from $h_0$. Thus a test configuration behaves like a vector field on the space of K\ahler potentials ${\cal H}$. We prove that $R$ is non-trivial if the ${\bf C}^\times$ action on $X_0$, the central fiber of $\cal X$, is non-trivial. The ray $R$ is obtained as limit of smooth geodesic rays $R_k\subset{\cal H}_k$, where ${\cal H}_k\subset{\cal H}$ is the subspace of Bergman metrics.

On the Limit Set of Discrete Subgroups of PU(2,1)

Let G be a discrete subgroup of PU(2,1); G acts on \(P^2_\mathbb C\) preserving the unit ball \(\mathbf H^2 _{\mathbb C}\), equipped with the Bergman metric. Let \(L(G) \subset S^3 = \partial \mathbf H^2 _{\mathbb C}\) be the limit set of G in the sense of Chen–Greenberg, and let\(\Lambda(G) \subset P^2_{\mathbb C}\) be the limit set of the G-action on\(P^2_{\mathbb C}\) in the sense of Kulkarni. We prove that L(G) = Λ(G) ∩ S 3 and Λ(G) is the union of all complex projective lines in \(P^2_\mathbb C\) which are tangent to S 3 at a point in L(G).

The Monge-Ampère operator and geodesics in the space of Kähler potentials

It is shown that geodesics in the space of K\"ahler potentials can be uniformly approximated by geodesics in the spaces of Bergman metrics. Two important tools in the proof are the Tian-Yau-Zelditch approximation theorem for K\"ahler potentials and the pluripotential theory of Bedford-Taylor, suitably adapted to K\"ahler manifolds.

The geometry of the Eisenstein-Picard modular group

The Eisenstein-Picard modular group PU(2,1;Z[ω]) is defined to be the subgroup of PU(2,1) whose entries lie in the ring Z[ω], where ω is a cube root of unity. This group acts isometrically and properly discontinuously on HC2, that is, on the unit ball in C2 with the Bergman metric. We construct a fundamental domain for the action of PU(2,1;Z[ω]) on HC2, which is a 4-simplex with one ideal vertex. As a consequence, we elicit a presentation of the group (see Theorem 5.9). This seems to be the simplest fundamental domain for a finite covolume subgroup of PU(2,1)