Brody Hyperbolic Manifold

A Brody hyperbolic manifold is a complex manifold for which every entire curve is constant. For compact , this condition is equivalent to nondegeneracy of the Kobayashi pseudodistance, which is constructed from chains of holomorphic disks. For example, a compact Riemann surface of genus at least 2 is Brody hyperbolic. Its universal cover is the unit disk, so an entire curve lifts to a bounded entire function and is constant by Liouville's theorem.

The compactness condition in the equivalence is essential. The condition is also distinct from a Kähler form becoming the exterior derivative of a bounded 1-form on the universal cover. The canonical-ampleness conjecture concerns Brody hyperbolicity together with compactness and a Kähler structure.