Leaves with isolated ends in foliated 3-manifolds

Abstract

RECENTLY, various authors have initiated a qualitative study of noncompact leaves in foliations of closed manifolds (e.g. [19,11,3, IS]). We continue this line of investigation for closed 3-manifolds foliated by surfaces. We will need the concept of ends of an open, connected manifold N[l, 17,111. These ends form a set 8(N) of ideal points at infinity in a compactication N U g(N) of the manifold. For instance, R has two ends ( 2 00) as does the cylinder S’ x R. The corresponding compactification of R is a closed interval [ 00, a] and that of S’ x R is a sphere S* with +w for the north pole and 00 for the south pole. More complicated examples are suggested in Fig. 1, where Nr has a sequence of isolated ends converging to one limit end, and N2 has isolated ends converging to first order limit ends, themselves converging to one second order limit end. Again the compactifications are homeomorphic to S*. As these examples suggest, 8(N) is always a compact, totally disconnected, separable space. The ends of a leaf L often become visible in terms of simpler leaves around which these ends are winding. There results a kind of Poincare-Bendixson theory for foliations of codimension one, of which we hope to give a more systematic account elsewhere. In this paper we restrict ourselves to leaves that are orientable surfaces and to ends that are isolated in ‘8(L), in which case the PoincarbBendixson theory is rather elementary. A leaf L in a compact manifold M has a well defined quasi-isomefry fype[l5]. That is, arbitrary Riemannian metrics relativized from A4 to L produce Riemannian structures on L that differ only up to bounded distortions. By now it is becoming clear that the question of what a leaf “looks like” should refer to the quasi-isometry type of L and not merely to its underlying