# Bergman metric

###### Definition.

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

 $g_{ij}(z):=\frac{\partial^{2}}{\partial z_{i}\partial\bar{z}_{j}}\log K(z,z),$

for $z\in G$. Then the length of a tangent vector $\xi\in T_{z}{\mathbb{C}}^{n}$ is then given by

 $\lvert\xi\rvert_{B,z}:=\sqrt{\sum_{i,j=1}^{n}g_{ij}(z)\xi_{i}\bar{\xi}_{j}}.$

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

 $\ell(\gamma)=\int_{0}^{1}\big{\lvert}\frac{\partial\gamma}{\partial t}(t)\big{% \rvert}_{B,\gamma(t)}dt.$

The distance $d_{G}(p,q)$ of two points $p,q\in G$ is then defined as

 $d_{G}(p,q):=\inf\{\ell(\gamma)\mid\text{ all piecewise C^{1} curves \gamma% such that \gamma(0)=p and \gamma(1)=q}\}.$

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 $G^{\prime}$. That is if $f$ is a biholomorphism of $G$ and $G^{\prime}$, then $d_{G}(p,q)=d_{G^{\prime}}(f(p),f(q))$.

## References

• 1 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Title Bergman metric BergmanMetric 2013-03-22 15:04:49 2013-03-22 15:04:49 jirka (4157) jirka (4157) 6 jirka (4157) Definition msc 32F45 BergmanKernel Bergman distance