---
tag: Y7W36GJDJD
kind: concept
name: Brody hyperbolicity
msc: [32Q45]
sources:
  - url: https://en.wikipedia.org/wiki/Kobayashi_metric
---

## Summary

A complex manifold is Brody hyperbolic if every holomorphic map from $\C$ into it is constant.
Nonconstant holomorphic maps $\C \to X$ are called entire curves, so Brody hyperbolicity means that $X$ contains no entire curve.
Every [[Kobayashi hyperbolic manifold|HRPAX8V7HA]] is Brody hyperbolic, and for compact manifolds the converse holds by [[Brody's lemma|VEV5E2077C]].
The notion is named after Robert Brody, who proved the converse in 1978.

## Explanations

### L1 human

Liouville's theorem says that a bounded holomorphic function on $\C$ is constant.
Brody hyperbolicity asks the same thing of a complex manifold: the only holomorphic maps from the whole complex plane into it are the constant maps.
The unit disc is Brody hyperbolic because of Liouville's theorem.
The plane minus two points is Brody hyperbolic by the little Picard theorem.
A complex torus $\C / \Lambda$ is not, because the quotient map $\C \to \C / \Lambda$ is a nonconstant holomorphic map.