Namely: every holomorphic map from D to itself is distance-decreasing with respect to the Poincaré metric on D. This is the beginning of a strong connection between complex analysis and the geometry of negative curvature.
[5] If a complex manifold X has a Hermitian metric with holomorphic sectional curvature bounded above by a negative constant, then X is Kobayashi hyperbolic.
The conjecture on hyperbolicity is known for hypersurfaces of high enough degree, thanks to a series of advances by Siu, Demailly and others, using the technique of jet differentials.
For example, Diverio, Merker and Rousseau showed that a general hypersurface in CPn+1 of degree at least 2n5 satisfies the Green-Griffiths-Lang conjecture.
McQuillan proved the Green–Griffiths–Lang conjecture for every complex projective surface of general type whose Chern numbers satisfy c12 > c2.
[12] For an arbitrary variety X of general type, Demailly showed that every holomorphic map C→ X satisfies some (in fact, many) algebraic differential equations.
[14] More generally, Campana gave a precise conjecture about which complex projective varieties X have Kobayashi pseudometric equal to zero.
In particular, let X be a projective variety over a number field k. Fix an embedding of k into C. Then Lang conjectured that the complex manifold X(C) is Kobayashi hyperbolic if and only if X has only finitely many F-rational points for every finite extension field F of k. This is consistent with the known results on rational points, notably Faltings's theorem on subvarieties of abelian varieties.
More precisely, let X be a projective variety of general type over a number field k. Let the exceptional set Y be the Zariski closure of the union of the images of all nonconstant holomorphic maps C → X.
[18] The Kobayashi–Eisenman pseudo-volume form is an intrinsic measure on a complex n-fold, based on holomorphic maps from the n-dimensional polydisc to X.
In particular, every projective variety of general type is measure-hyperbolic, meaning that the Kobayashi–Eisenman pseudo-volume form is positive outside a lower-dimensional algebraic subset.