Geometric aspects of transversal Killing spinors on Riemannian flows

We study a Killing spinor type equation on spin Riemannian flows. We prove integrability conditions and partially classify those flows carrying non-trivial solutions.      

Hopf cyclic cohomology and transverse characteristic classes

We refine the cyclic cohomological apparatus for computing the Hopf cyclic cohomology of the Hopf algebras associated to infinite primitive Cartan–Lie pseudogroups, and for the transfer of their characteristic classes to foliations. The main novel feature is the precise identification as a Hopf cyclic complex of the image of the canonical homomorphism from the Gelfand–Fuks complex to the Bott complex for equivariant cohomology. This provides a convenient new model for the Hopf cyclic cohomology of the geometric Hopf algebras, which allows for an efficient transport of the Hopf cyclic classes via characteristic homomorphisms. We illustrate the latter aspect by indicating how to realize the universal Hopf cyclic Chern classes in terms of explicit cocycles in the cyclic cohomology of étale foliation groupoids.     

Geometric non-commutative geometry

In a recent paper, the authors proved that no spin foliation on a compact enlargeable manifold with Hausdorff homotopy graph admits a metric of positive scalar curvature on its leaves. This result extends groundbreaking results of Lichnerowicz, Gromov and Lawson, and Connes on the non-existence of metrics of positive scalar curvature. In this paper we review in more detail the material needed for the proof of our theorem and we extend our non-existence results to non-compact manifolds of bounded geometry. We also give a first obstruction result for the existence of metrics with (not necessarily uniform) leafwise PSC in terms of the A-hat class in Haefliger cohomology.     

Smectic Liquid Crystals: Materials with One-dimensional, Periodic Order

Smectic liquid crystals are materials formed by stacking deformable, fluid layers. Although smectics prefer to have flat, uniformly-spaced layers, boundary conditions can impose curvature on the layers. Since the layer spacing and curvature are intertwined, the problem of finding minimal configurations for the layers becomes nontrivial. We discuss various topological and geometrical aspects of these materials and present recent progress on finding some exact layer configurations. We also exhibit connections to the study of certain embedded minimal surfaces and briefly summarize some important open problems.     

Twisted longitudinal index theorem for foliations and wrong way functoriality

For a Lie groupoid G with a twisting σ (a PU(H)-principal bundle over G), we use the (geometric) deformation quantization techniques supplied by Connes tangent groupoids to define an analytic index morphism in twisted K-theory. In the case the twisting is trivial we recover the analytic index morphism of the groupoid.For a smooth foliated manifold with twistings on the holonomy groupoid we prove the twisted analog of the Connes–Skandalis longitudinal index theorem. When the foliation is given by fibers of a fibration, our index coincides with the one recently introduced by Mathai, Melrose, and Singer.We construct the pushforward map in twisted K-theory associated to any smooth (generalized) map f:W→M/F and a twisting σ on the holonomy groupoid M/F, next we use the longitudinal index theorem to prove the functoriality of this construction. We generalize in this way the wrong way functoriality results of Connes and Skandalis when the twisting is trivial and of Carey and Wang for manifolds.     

On smooth foliations with Morse singularities

Let M be a smooth manifold and let F be a codimension one, C^∞ foliation on M, with isolated singularities of Morse type. The study and classification of pairs (M,F) is a challenging (and difficult) problem. In this setting, a classical result due to Reeb (1946) [11] states that a manifold admitting a foliation with exactly two center-type singularities is a sphere. In particular this is true if the foliation is given by a function. Along these lines a result due to Eells and Kuiper (1962) [4] classifies manifolds having a real-valued function admitting exactly three non-degenerate singular points. In the present paper, we prove a generalization of the above mentioned results. To do this, we first describe the possible arrangements of pairs of singularities and the corresponding codimension one invariant sets, and then we give an elimination procedure for suitable center-saddle and some saddle-saddle configurations (of consecutive indices).In the second part, we investigate if other classical results, such as Haefliger and Novikov (Compact Leaf) theorems, proved for regular foliations, still hold true in presence of singularities. At this purpose, in the singular set, Sing(F) of the foliation F, we consider weakly stable components, that we define as those components admitting a neighborhood where all leaves are compact. If Sing(F) admits only weakly stable components, given by smoothly embedded curves diffeomorphic to S¹, we are able to extend Haefliger?s theorem. Finally, the existence of a closed curve, transverse to the foliation, leads us to state a Novikov-type result.     

Semiclassical Spectral Asymptotics on Foliated Manifolds

We consider a (hypo)elliptic pseudodifferential operator A_h on a closed foliated manifold (M,ℱ), depending on a parameterh > 0, of the form A_h = A+h^mB, where A is a formally self–adjoint tangentially elliptic operator of orderμ > 0 with the nonnegative principal symbol and B is a formally self–adjoint classical pseudodi.erential operator of orderm > 0 on M with the holonomy invariant transversal principal symbol such that its principal symbol is positive, if μ < m, and its transversal principal symbol is positive, if μ ≥ m. We prove an asymptotic formula for the eigenvalue distribution function N_h(λ) of the operator A_h when h tends to 0 and λ is constant.     

Regularity of volume-minimizing flows on 3-manifolds

In Johnson and Smith (Indiana Univ Math J 44:45–85, 1995; Ann Global Anal Geometry 30:239–287, 2006; Proceedings of the VII International Colloquium on Differential Geometry, 1994, World Scientific, pp. 81–98), the authors characterized the singular set (discontinuities of the graph) of a volume-minimizing rectifiable section of a fiber bundle, showing that, except under certain circumstances, there exists a volume-minimizing rectifiable section with the singular set lying over a codimension-3 set in the base space. In particular, it was shown that for 2-sphere bundles over 3-manifolds, a minimizer exists with a discrete set of singular points. In this article, we show that for a 2-sphere bundle over a compact 3-manifold, such a singular point cannot exist. As a corollary, for any compact 3-manifold, there is a C ¹ volume-minimizing one-dimensional foliation. In addition, this same analysis is used to show that the examples, due to Pedersen (Trans Am Math Soc 336:69–78, 1993), of potentially volume-minimizing rectifiable sections (rectifiable foliations) of the unit tangent bundle to S ²ⁿ⁺¹ are not, in fact, volume minimizing.      

Hyperbolic ends with particles and grafting on singular surfaces

We prove that any 3-dimensional hyperbolic end with particles (cone singularities along infinite curves of angles less than π) admits a unique foliation by constant Gauss curvature surfaces. Using a form of duality between hyperbolic ends with particles and convex globally hyperbolic maximal (GHM) de Sitter spacetime with particles, it follows that any 3-dimensional convex GHM de Sitter spacetime with particles also admits a unique foliation by constant Gauss curvature surfaces. We prove that the grafting map from the product of Teichmüller space with the space of measured laminations to the space of complex projective structures is a homeomorphism for surfaces with cone singularities of angles less than π, as well as an analogue when grafting is replaced by “smooth grafting”.     

Generic positivity and foliations in positive characteristic

We prove generic semipositivity of the tangent bundle of a non-uniruled Calabi–Yau variety in positive characteristic. We also construct an example of a nef line bundle in characteristic zero, whose each reduction to positive characteristic is not nef.