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Bergman metric
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(Definition)
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Definition 1 Let $G \subset {\mathbb{C}}^n$ be a domain and let $K(z,w)$ be the Bergman kernel on $G$ We define a Hermitian metric on the tangent bundle $T_z {\mathbb{C}}^n$ by \begin{equation*} g_{ij} (z) := \frac{\partial^2}{\partial z_i \partial \bar{z}_j} \log K(z,z) , \end{equation*}for $z \in G$ Then
the length of a tangent vector $\xi \in T_z{\mathbb{C}}^n$ is then given by \begin{equation*} \lvert \xi \rvert_{B,z} := \sqrt{\sum_{i,j=1}^n g_{ij}(z) \xi_i \bar{\xi}_j }. \end{equation*}This metric is called the Bergman metric on $G$
The length of a (piecewise) $C^1$ curve $\gamma \colon [0,1] \to {\mathbb{C}}^n$ is then computed as \begin{equation*} \ell (\gamma) = \int_0^1 \big\lvert \frac{\partial \gamma}{\partial t}(t) \big\rvert_{B,\gamma(t)} dt . \end{equation*}The distance $d_G(p,q)$ of two points $p,q \in G$ is then defined as \begin{equation*} d_G(p,q):= \inf \{ \ell (\gamma) \mid \text{ all piecewise $C^1$ curves $\gamma$ such that $\gamma(0)=p$ and $\gamma(1)=q$} \} .
\end{equation*}The distance $d_G$ is called the Bergman distance.
The Bergman metric is in fact a positive definite matrix at each point if $G$ is a bounded domain. More importantly, the distance $d_G$ is invariant under biholomorphic mappings of $G$ to another domain $G'$ That is if $f$ is a biholomorphism of $G$ and $G'$ then $d_G(p,q) =
d_{G'}(f(p),f(q))$
- 1
- Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.
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"Bergman metric" is owned by jirka.
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Cross-references: biholomorphic mappings, invariant, bounded, matrix, positive definite, points, distance, curve, piecewise, tangent vector, length, tangent bundle, metric, Hermitian, Bergman kernel, domain
This is version 3 of Bergman metric, born on 2005-02-22, modified 2005-03-05.
Object id is 6803, canonical name is BergmanMetric.
Accessed 3285 times total.
Classification:
| AMS MSC: | 32F45 (Several complex variables and analytic spaces :: Geometric convexity :: Invariant metrics and pseudodistances) |
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Pending Errata and Addenda
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