Bergman metric

In differential geometry, the Bergman metric is a Hermitian metric that can be defined on certain types of complex manifold. It is so called because it is derived from the Bergman kernel, both of which are named for Stefan Bergman.

Definition

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

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

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

${\displaystyle \left\vert \xi \right\vert _{B,z}:={\sqrt {\sum _{i,j=1}^{n}g_{ij}(z)\xi _{i}{\bar {\xi }}_{j}}}.}$

This metric is called the Bergman metric on G.

The length of a (piecewise) C¹ curve ${\displaystyle \gamma \colon [0,1]\to {\mathbb {C} }^{n}}$ is then computed as

${\displaystyle \ell (\gamma )=\int _{0}^{1}\left\vert {\frac {\partial \gamma }{\partial t}}(t)\right\vert _{B,\gamma (t)}dt.}$

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

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

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

References

• Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.

This article incorporates material from Bergman metric on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.