Summary: HYPERBOLICITY IN UNBOUNDED CONVEX DOMAINS
FILIPPO BRACCI AND ALBERTO SARACCO
ABSTRACT. We provide several equivalent characterizations of Kobayashi hyperbolicity in un-
bounded convexdomains in terms of peak and anti-peak functions at infinity, affine lines, Bergman
metric and iteration theory.
Despite the fact that linear convexity is not an invariant property in complex analysis, bounded
convex domains in CN
have been very much studied as prototypes for the general situation.
In particular, by Harris' theorem  (see also, , ) it is known that bounded convex
domains are always Kobayashi complete hyperbolic (and thus by Royden's theorem, they are
also taut and hyperbolic). Moreover, by Lempert's theorem , , the Kobayashi distance
can be realized by means of extremal discs. These are the basic cornerstone for many useful
results, especially in pluripotential theory and iteration theory.
On the other hand, not much is known about unbounded domains. Clearly, the geometry at
infinity must play some important role. In this direction, Gaussier  gave some conditions
in terms of existence of peak and anti-peak functions at infinity for an unbounded domain to
be hyperbolic, taut or complete hyperbolic. Recently, Nikolov and Pflug  deeply studied
conditions at infinity which guarantee hyperbolicity, up to a characterization of hyperbolicity in
terms of the asymptotic behavior of the Lempert function.