TITLE: Does $\mathbb{CP}^2$ admit a Riemann surface lamination structure?
QUESTION [9 upvotes]: Does $\mathbb{CP}^2$ admit a Riemann surface lamination structure?  Every paper or article I looked at, talk only about singular laminations on $\mathbb{CP}^2$.  I was wondering why.  If you know something about it or you can give some reference, it would be nice.

REPLY [11 votes]: It is conjectured that $\mathbb {CP}^2$ contains no embedded compact laminated set (without singularities) apart the smooth algebraic curves.
This is a strong form of the "Minimal Exceptional" conjecture, stating that for a singular holomorphic foliation of $\mathbb{CP}^2$, every leaf accumulates in the singular set.
Nice references about this subject are the following surveys:

É. Ghys - Laminations par surfaces de Riemann. Dynamique et géométrie complexes -
Panor. Synthèses, 8, 1999.
S. Zakeri - Dynamics of singular holomorphic foliations on the complex projective plane. Laminations and foliations in dynamics, geometry and topology -
Contemp. Math., 269, Amer. Math. Soc., 2001.
J. E. Fornaess and N. Sibony - Riemann surface laminations with singularities. - J. Geom. Anal. 18 (2008), no. 2, 400–442.