Logarithmic Surfaces and Hyperbolicity
  In 1981 J.Noguchi proved that in a logarithmic algebraic manifold, having logarithmic irregularity strictly bigger than its dimension, any entire curve is algebraically degenerate.   In the present paper we are interested in the case of manifolds having logarithmic irregularity equal to its dimension. We restrict our attention to Brody curves, for which we resolve the problem completely in dimension 2:   Theorem: In a logarithmic surface with logarithmic irregularity 2 and logarithmic Kodaira dimension 2, any Brody curve is algebraically degenerate. We also deal with the case of arbitrary logarithmic Kodaira dimension.   As a corollary, we get hyperbolicity for such logarithmic surfaces not containing non-hyperbolic algebraic curves and having hyperbolically stratified boundary divisors. In particular we get the "best possible" result on algebraic degeneracy of Brody curves in the complex plane minus a curve consisting of three components, thus improving results of Dethloff-Schumacher-Wong from 1995. 
Gerd Dethloff and Steven Lu