On the existence of non-trivial laminations in ${\mathbb{CP}}^2$

Abstract

In this article, we show the existence of a nontrivial Riemann surface lamination embedded in ${\mathbb{CP}}^2$ by using Donaldson's construction of asymptotically holomorphic submanifolds. Further, the lamination we obtain has the property that each leaf is a totally geodesic submanifold of ${\mathbb{CP}}^2$ with respect to the Fubini-Study metric. This may constitute a step in understanding the conjecture on the existence of minimal exceptional sets in ${\mathbb{CP}}^2$.

Caution: There is an error in the argument (thanks to E. Ghys for pointing it out). We believe that it can be fixed and article will be updated soon with a correct argument. We thank you for your patience