Existence of codimension one foliations with minimal leaves

Every smooth closed manifold of dimension 4 or greater that has a smooth codimension one foliation, has such aC ¹ foliation whose leaves are minimal hypersurfaces for someC ¹ Riemannian metric.     

The singularities of the wave trace of the basic Laplacian of a Riemannian foliation

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 distributions and demonstrate the existence of the basic wave kernel. The singularities of the trace of this basic wave kernel occur at the lengths of certain geodesic arcs which are orthogonal to the closures of the leaves of the foliation. In cases when the foliation has regular closure, a complete representation of the trace of the basic wave kernel can be computed for t≠0. Otherwise, a partial trace formula over a certain set of lengths of well-behaved geodesic arcs is obtained.     

Extrinsic Geometric Flows on Codimension-One Foliations

We study the geometry of a codimension-one foliation with a time-dependent Riemannian metric. The work begins with formulae for deformations of geometric quantities as the Riemannian metric varies along the leaves of a foliation. Then the Extrinsic Geometric Flow depending on the second fundamental form of the foliation is introduced. Under suitable assumptions, this evolution yields the second-order parabolic PDEs, for which the existence/uniqueness and in some cases convergence of a solution are shown. Applications to the problem of prescribing the mean curvature function of a codimension-one foliation, and examples with harmonic and umbilical foliations (e.g., foliated surfaces) and with twisted product metrics are given.     

On the structure of a Morse form foliation

The foliation of a Morse form ω on a closed manifold M is considered. Its maximal components (cylinders formed by compact leaves) form the foliation graph; the cycle rank of this graph is calculated. The number of minimal and maximal components is estimated in terms of characteristics of M and ω. Conditions for the presence of minimal components and homologically non-trivial compact leaves are given in terms of rk ω and Sing ω. The set of the ranks of all forms defining a given foliation without minimal components is described. It is shown that if ω has more centers than conic singularities then b ₁(M) = 0 and thus the foliation has no minimal components and homologically non-trivial compact leaves, its folitation graph being a tree.      

A Note on the Linearity of Real-valued Functions with Respect to Suitable Metrics

In this paper we prove that for every real-valued Morse function φ on a smooth closed manifold ℳ and every neighborhood U of its critical points a suitable Riemannian metric μ _U exists such that φ is linear outside U     

Hodge cohomology of some foliated boundary and foliated cusp metrics

For fibred boundary and fibred cusp metrics, Hausel, Hunsicker, and Mazzeo identified the space of L² harmonic forms of fixed degree with the images of maps between intersection cohomology groups of an associated stratified space obtained by collapsing the fibres of the fibration at infinity onto its base. In the present paper, we obtain a generalization of this result to situations where, rather than a fibration at infinity, there is a Riemannian foliation with compact leaves admitting a resolution by a fibration. If the associated stratified space (obtained now by collapsing the leaves of the foliation) is a Witt space and if the metric considered is a foliated cusp metric, then no such resolution is required.     

Long Time Behaviour of Leafwise Heat Flow for Riemannian Foliations

For any Riemannian foliation F on a closed manifold M with an arbitrary bundle-like metric, leafwise heat flow of differential forms is proved to preserve smoothness on M at infinite time. This result and its proof have consequences about the space of bundle-like metrics on M, about the dimension of the space of leafwise harmonic forms, and mainly about the second term of the differentiable spectral sequence of F.     

Minimal singular Riemannian foliations

We prove that a singular foliation on a compact manifold admitting an adapted Riemannian metric for which all leaves are minimal must be regular. To cite this article: V. Miquel, R.A. Wolak, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Closed forms transverse to singular foliations

We describe a homological criterion for the existence of a closed form transverse to a foliation which is allowed to have certain tame singularities, generalizing results in the nonsingular case by Sullivan. As illustrations of the method, we derive results about the existence of singular symplectic forms and about the characterization of intrinsic harmonicity for forms.     

Attractors and an analog of the Lichnérowicz conjecture for conformal foliations

We prove that each codimension q ≥ 3 conformal foliation (M,F) either is Riemannian or has a minimal set that is an attractor. If (M,F) is a proper conformal foliation that is not Riemannian then there exists a closed leaf that is an attractor. We do not assume that M is compact. Moreover, if M is compact then a non-Riemannian conformal foliation (M,F) is a (Conf(S ^q ), S ^q )-foliation with a finite family of attractors, and each leaf of this foliation belongs to the basin of at least one attractor.     

Higher order curvature flows on surfaces

We consider a sixth and an eighth order conformal flow on Riemannian surfaces, which arise as gradient flows for the Calabi energy with respect to a higher order metric. Motivated by a work of Struwe which unified the approach to the Hamilton-Ricci and Calabi flow, we extend the method to these higher order cases. Our results contain global existence and exponentially fast convergence to a constant scalar curvature metric.Uniform bounds on the conformal factor are obtained via the concentration-compactness result for conformal metrics. In the case of the sphere we use the idea of DeTurck's gauge flow to derive bounds up to conformal transformation.We prove exponential convergence by showing that the Calabi energy decreases exponentially fast. The problem of the non-trivial kernel in the evolution of the Calabi energy on the sphere is resolved by using Kazdan-Warner's identity. Mathematics subject classifications (2000) {Primary 53C44 Secondary 35K25}     

The center foliation of an affine diffeomorphism

Given an affine (i.e. connection-preserving) diffeomorphismf of a Riemannian manifoldM, we consider itscenter foliation, N, comprised by the directions that neither expand nor contract exponentially under the action generated byf. The main remarks made here (Corollary 3 and Theorem 7) are: There exists a metric compatible with the Levi-Civita connection for which the universal cover ofM decomposes isometrically as the Riemannian product of the universal cover of a leaf ofN (these covers are all isometric) and the Euclidean space; and ifN is one-dimensional,M is flat and the foliation is (up to finite cover) the fiber foliation of a Riemannian submersion onto a flat torus.     

Extremal Metrics for Quadratic Functional of Scalar Curvature on Closed 3-Manifolds

In this paper, we first show the global existence of the three-dimensionalCalabi flow on any closed 3-manifold with an arbitrary background metric g ₀. Second, we show the asymptotic convergence of a subsequence ofsolutions of the Calabi flow on a closed 3-manifold with Yamabe constant Q < 0 or Q = 0 and Q > 0, up to conformal transformations. With itsapplication, we prove the existence of extremal metrics for quadraticfunctional of scalar curvature on a closed 3-manifold which is served asan extension of the Yamabe problem on closed manifolds. Moreover, theexistence of extremal metrics on complete noncompact 3-manifolds willdiscuss elsewhere.     

CMC Foliations of Closed Manifolds

We prove that every closed, smooth \(n\)-manifold \(X\) admits a Riemannian metric together with a constant mean curvature (CMC) foliation if and only if its Euler characteristic is zero, where by a CMC foliation we mean a smooth, codimension-one, transversely oriented foliation with leaves of CMC and where the value of the CMC can vary from leaf to leaf. Furthermore, we prove that this CMC foliation of \(X\) can be chosen so that when \(n\ge 2\), the constant values of the mean curvatures of its leaves change sign. We also prove a general structure theorem for any such non-minimal CMC foliation of \(X\) that describes relationships between the geometry and topology of the leaves, including the property that there exist compact leaves for every attained value of the mean curvature.     

The Calabi metric for the space of Kähler metrics

Given any closed Kähler manifold we define, following an idea by Calabi (Bull. Am. Math. Soc. 60:167–168, 1954), a Riemannian metric on the space of Kähler metrics regarded as an infinite dimensional manifold. We prove several geometrical features of the resulting space, some of which we think were already known to Calabi. In particular, the space is a portion of an infinite dimensional sphere and explicit unique smooth solutions for the Cauchy and the Dirichlet problems for the geodesic equation are given.     

Existence and regularity of maximal metrics for the first Laplace eigenvalue on surfaces

We investigate in this paper the existence of a metric which maximizes the first eigenvalue of the Laplacian on Riemannian surfaces. We first prove that, in a given conformal class, there always exists such a maximizing metric which is smooth except at a finite set of conical singularities. This result is similar to the beautiful result concerning Steklov eigenvalues recently obtained by Fraser and Schoen (Sharp eigenvalue bounds and minimal surfaces in the ball, 2013). Then we get existence results among all metrics on surfaces of a given genus, leading to the existence of minimal isometric immersions of smooth compact Riemannian manifold (M, g) of dimension 2 into some k-sphere by first eigenfunctions. At last, we also answer a conjecture of Friedlander and Nadirashvili (Int Math Res Not 17:939–952, 1999) which asserts that the supremum of the first eigenvalue of the Laplacian on a conformal class can be taken as close as we want of its value on the sphere on any orientable surface.     

Partial Harmonicity of Continuous Maximal Plurisubharmonic Functions

If u is a sufficiently smooth maximal plurisubharmonic function such that the complex Hessian of u has constant rank, it is known that there exists a foliation by complex manifolds, such that u is harmonic along the leaves of the foliation. In this paper, we show a partial analogue of this result for maximal plurisubharmonic functions that are merely continuous, without the assumption on the complex Hessian. In this setting, we cannot expect a foliation by complex manifolds, but we prove the existence of positive currents of bidimension (1, 1) such that the function is harmonic along the currents.     

On foliated circle bundles over closed orientable 3-manifolds

We show that there exists a family of smooth orientable circle bundles over closed orientable 3-manifolds each of which has a codimension-one foliation transverse to the fibres of class C ⁰ but has none of class C ³ . There arises a necessary condition induced from the Milnor-Wood inequality for the existence of a foliation transverse to the fibres of an orientable circle bundle over a closed orientable 3-manifold. We show that with some exceptions this necessary condition is also sufficient for the existence of a smooth transverse foliation if the base space is a closed Seifert fibred manifold. Received: May 13, 1996     

Proofs of Conjectures about Singular Riemannian Foliations

A singular foliation on a complete Riemannian manifold M is said to be Riemannian if each geodesic that is perpendicular at one point to a leaf remains perpendicular to every leaf it meets. We prove that if the distribution of normal spaces to the regular leaves is integrable, then each leaf of this normal distribution can be extended to be a complete immersed totally geodesic submanifold (called section), which meets every leaf orthogonally. In addition the set of regular points is open and dense in each section. This result generalizes a result of Boualem and solves a problem inspired by a remark of Palais and Terng and a work of Szenthe about polar actions. We also study the singular holonomy of a singular Riemannian foliation with sections (s.r.f.s. for short) and in particular the tranverse orbit of the closure of each leaf. Furthermore we prove that the closures of the leaves of a s.r.f.s on M form a partition of M which is a singular Riemannian foliation. This result proves partially a conjecture of Molino.