# Bergman metric

###### Definition.

Let $G\subset {\u2102}^{n}$ be a domain and let $K(z,w)$ be the Bergman kernel on $G$. We define a Hermitian metric on the tangent bundle ${T}_{z}{\u2102}^{n}$ by

$${g}_{ij}(z):=\frac{{\partial}^{2}}{\partial {z}_{i}\partial {\overline{z}}_{j}}\mathrm{log}K(z,z),$$ |

for $z\in G$. Then the length of a tangent vector $\xi \in {T}_{z}{\u2102}^{n}$ is then given by

$${|\xi |}_{B,z}:=\sqrt{\sum _{i,j=1}^{n}{g}_{ij}(z){\xi}_{i}{\overline{\xi}}_{j}}.$$ |

This metric is called the Bergman metric on $G$.

The length of a (piecewise) ${C}^{1}$ curve $\gamma :[0,1]\to {\u2102}^{n}$ is then computed as

$$\mathrm{\ell}(\gamma )={\int}_{0}^{1}{\left|\frac{\partial \gamma}{\partial t}(t)\right|}_{B,\gamma (t)}\mathit{d}t.$$ |

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

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

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 |
---|---|

Canonical name | BergmanMetric |

Date of creation | 2013-03-22 15:04:49 |

Last modified on | 2013-03-22 15:04:49 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 6 |

Author | jirka (4157) |

Entry type | Definition |

Classification | msc 32F45 |

Related topic | BergmanKernel |

Defines | Bergman distance |