Planar surfaces in three-manifolds

In this paper we consider the problem of algorithmic finding of a proper essential planar surface in a given irreducible orientable compact $3$-manifold. The method uses the Haken theory of normal surfaces in $3$-manifolds with boundary pattern [5]. The solution is based on an estimate, considered in [3], of the average length of boundary curves of an arbitrary proper essential planar surface and on the notion of a slope in the boundary of an arbitrary manifold that generalizes the notion of a slope on a torus.