D.E . BARRETT AND J . E . FORNAESS We study the regularity of the induced foliation of a Levi-flat hypersurface in C'° , showing that the foliation is as many times continuously differentiable as the hypersurface itself. The key step in the proof given here is the construction of a certain family of approximate plurisubharmonic defining functions for the hypersurface in question .

Let F be an oriented topological foliation on S = {(x, y, z) ∈ R;x + y + z = 1} having only a finite number of singularities. If F has only a finite number of closed orbits and satisfies one additional condition, then it is shown that F is topologically equivalent to (the foliation induced by) a polynomial vector field.

The pivotal topic of this paper is the study of Levi-flat real hypersurfaces S with circular fibers in a rank 1 affine bundle A over a Riemann surface X. (To say that S is Levi-flat is to say that S admits a foliation by Riemann surfaces; equivalently, in the language of [SuTh], S may be said to prescribe a holomorphic motion of circles through A.) After setting notation and terminology in §2 w...

In this paper we study foliations determined by a closed 1-form with Morse singularities on smooth compact manifolds. The problem of the topological structure of the level surfaces of such forms was posed by Novikov in [1], and has been studied in [2]-[5]. In the present article we investigate the problem of the existence of a non-compact leaf, verify a test for non-compactness of a foliation i...

We apply techniques of microlocal analysis to the study of the transverse geometry of Riemannian foliations in order to analyze spectral invariants of the basic Laplacian acting on functions on a Riemannian foliation with a bundle-like metric. In particular, we consider the trace of the basic wave operator when the mean curvature form is basic. We extend the concept of basic functions to distri...

In this work, we develop shape expansions of minimal matchbox manifolds without holonomy, in terms of branched manifolds formed from their leaves. Our approach is based on the method of coding the holonomy groups for the foliated spaces, to define leafwise regions which are transversely stable and are adapted to the foliation dynamics. Approximations are obtained by collapsing appropriately cho...

The concept of time is discussed in the context of the canonical formulation of the gravitational field. Using a hypersurface orthogonal foliation, the arbitrariness of the lapse function is eliminated and the shift vector vanishes, allowing a consistent definition of time. I. ON LAPSES AND SHIFTS As it well know, the space-time of Newtonian mechanics is foliated by globally defined 3-dimension...

Biconformal deformations take place in the presence of a conformal foliation, deforming by different factors tangent to and orthogonal foliation. Four-manifolds endowed with foliation surfaces present natural context put into effect this process. We develop tools calculate transformation Ricci curvature under such apply our method construct Einstein $4$-manifolds. One particular family examples...