Bergman metric

Definition.

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

$$g_{ij}(z):=\frac{{\partial }^2}{\partial z_i\partial {\overline{z}}_j}\log K(z,z),$$

for $z\in G$. Then the length of a tangent vector $\xi \in T_z{ℂ}^n$ is thengiven 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 $\gamma :[0,1]\to {ℂ}^n$ isthen computed as

$$\mathrm{\ell }(\gamma )={\int }_0^1{\left|\frac{\partial \gamma }{\partial t}(t)\right|}_{B,\gamma (t)}𝑑t.$$

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

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

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