The holomorphic Liouville property of a connected complex manifold means that every bounded holomorphic
function on it is constant. Writing 
for these functions,
the condition is 
. Complex Euclidean
space has this property by Liouville's
boundedness theorem. A bounded domain has nonconstant
bounded coordinate functions and therefore fails it.
Holomorphic Liouville Property
See also
Holomorphic Function, Holomorphic Map, Kähler Metric, Liouville's Boundedness Theorem, Sectional CurvatureExplore with Wolfram|Alpha
References
Lin, V. Ya. and Zaidenberg, M. "Liouville and Carathéodory Coverings in Riemannian and Complex Geometry." In Voronezh Winter Mathematical Schools: Dedicated to Selim Krein (Ed. P. Kuchment and V. Lin). Providence, RI: Amer. Math. Soc., pp. 111-130, 1998. https://doi.org/10.1090/trans2/184/08.Cite this as:
Weisstein, Eric W. "Holomorphic Liouville Property." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HolomorphicLiouvilleProperty.html