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

In this article, we show the existence of a nontrivial Riemann surface lamination embedded in ${\mathbb{C}\mathbb{P}}^2$ 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 ${\mathbb{C}\mathbb{P}}^2$ with respect to the Fubini-Study metric. This may constitute a step in understanding the conjecture on the existence of minimal exceptional sets in ${\mathbb{C}\mathbb{P}}^2$.