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

TL;DR

This paper demonstrates the existence of a non-trivial Riemann surface lamination in complex projective plane using asymptotically holomorphic techniques, with leaves that are totally geodesic, advancing understanding of minimal sets.

Contribution

It constructs a non-trivial lamination in $\,\mathbb{CP}^2$ with totally geodesic leaves, utilizing Donaldson's asymptotically holomorphic methods, which is a novel approach.

Findings

Existence of a non-trivial lamination in $\,\mathbb{CP}^2$

Leaves are totally geodesic submanifolds

Potential implications for minimal exceptional sets

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 $\mathbb{CP}^2$.