Existence of dicritical singularities of Levi-flat hypersurfaces and holomorphic foliations

We study holomorphic foliations tangent to singular real-analytic Levi-flat hypersurfaces in compact complex manifolds of complex dimension two. We give some hypotheses to guarantee the existence of dicritical singularities of these objects. As consequence, we give some applications to holomorphic foliations tangent to real-analytic Levi-flat hypersurfaces with singularities in \(\mathbb {P}^2\).     

Nowhere minimal CR submanifolds and Levi-flat hypersurfaces

A local uniqueness property of holomorphic functions on real-analytic nowhere minimal CR submanifolds of higher codimension is investigated. A sufficient condition called almost minimality is given and studied. A weaker necessary condition, being contained a possibly singular real-analytic Levi-flat hypersurface is studied and characterized. This question is completely resolved for algebraic submanifolds of codimension 2 and a sufficient condition for noncontainment is given for non algebraic submanifolds. As a consequence, an example of a submanifold of codimension 2, not biholomorphically equivalent to an algebraic one, is given. We also investigate the structure of singularities of Levi-flat hypersurfaces.     

Singular set of a Levi-flat hypersurface is Levi-flat

We study the singular set of a singular Levi-flat real-analytic hypersurface. We prove that the singular set of such a hypersurface is Levi-flat in the appropriate sense. We also show that if the singular set is small enough, then the Levi-foliation extends to a singular codimension one holomorphic foliation of a neighborhood of the hypersurface.     

On Levi-Flat Hypersurfaces with Generic Real Singular Set

In this paper, we consider the holomorphic extension problem for the Levi foliation of a Levi-flat real-analytic hypersurface whose singular set is a generic real submanifold.     

Algebraic Levi-Flat Hypervarieties in Complex Projective Space

We study singular real-analytic Levi-flat hypersurfaces in complex projective space. We define the rank of an algebraic Levi-flat hypersurface and study the connections between rank, degree, and the type and size of the singularity. In particular, we study degenerate singularities of algebraic Levi-flat hypersurfaces. We then give necessary and sufficient conditions for a Levi-flat hypersurface to be a pullback of a real-analytic curve in ℂ via a meromorphic function. Among other examples, we construct a nonalgebraic semianalytic Levi-flat hypersurface with compact leaves that is a perturbation of an algebraic Levi-flat variety.     

A new geometric construction of compact complex manifolds in any dimension

We consider holomorphic linear foliations of dimension m of (with ) fulfilling a so-called weak hyperbolicity condition and equip the projectivization of the leaf space (for the foliation restricted to an adequate open dense subset) with a structure of compact, complex manifold of dimension . We show that, except for the limit-case where we obtain any complex torus of any dimension, this construction gives non-symplectic manifolds, including the previous examples of Hopf, Calabi-Eckmann, Haefliger (linear case), Loeb-Nicolau (linear case) and López de Medrano-Verjovsky. We study some properties of these manifolds, that is to say meromorphic functions, holomorphic vector fields, forms and submanifolds. For each manifold, we construct an analytic space of deformations of dimension and show that, under some additional conditions, it is universal. Lastly, we give explicit examples of new compact, complex manifolds, in particular of connected sums of products of spheres and show the existence of a momentum-like map which classifies these manifolds, up to diffeomorphism. Received: 28 October 1998 / in final form: 7 September 1999     

Diffeomorphisms preserving R‐circles in three dimensional CR manifolds

R ‐circles in (non‐degenerate) three dimensional CR manifolds are the analogues to traces of Lagrangian totally geodesic planes on S³ viewed as the boundary of two dimensional complex hyperbolic space. They form a family of certain Legendrian curves on the manifold. We prove that a diffeomorphism between three dimensional CR manifolds which preserve circles is either a CR diffeomorphism or a conjugate CR diffeomorphism.     

Harmonic Riemannian maps on locally conformal Kaehler manifolds

We study harmonic Riemannian maps on locally conformal Kaehler manifolds (lcK manifolds). We show that if a Riemannian holomorphic map between lcK manifolds is harmonic, then the Lee vector field of the domain belongs to the kernel of the Riemannian map under a condition. When the domain is Kaehler, we prove that a Riemannian holomorphic map is harmonic if and only if the lcK manifold is Kaehler. Then we find similar results for Riemannian maps between lcK manifolds and Sasakian manifolds. Finally, we check the constancy of some maps between almost complex (or almost contact) manifolds and almost product manifolds.     

Levi-Flat Hypersurfaces Tangent to Projective Foliations

Let M?? ⁿ be a singular real-analytic Levi-flat hypersurface tangent to a codimension-one holomorphic foliation \(\mathcal{F}\) on ? ⁿ . For n≥3, we give sufficient conditions to guarantee the existence of degenerate singularities in M, (in the sense of Segre varieties) and as a consequence we prove that \(\mathcal{F}\) can be defined by a global closed meromorphic 1-form.     

Modelling by a planar map: Lyapunov-type numbers in the presence of spiral points

Lyapunov-type numbers are usually defined for diffeomorphisms with a smooth invariant manifold. We consider here the case of a planar diffeomorphism with an invariant curve that contains spiral points. The limits defining the Lyapunov-type numbers are shown to exist. Numerical results for the delayed logistic map illustrate the analysis.     

On the L 2-extension of twisted holomorphic sections of singular hermitian line bundles

We prove a sharp Ohsawa–Takegoshi–Manivel type L ²-extension result for twisted holomorphic sections of singular hermitian line bundles over almost Stein manifolds. We establish as corollaries some extension results for pluri-twisted holomorphic sections of singular hermitian line bundles over projective manifolds.     

Approximation of Singular Discs for CR Extension

We introduce a class of singular discs, modelled on the one introduced by Zaitsev, Zampieri and the author. We show that they gain regularity from reparametrization and from taking powers. We set up a Bishop equation to attach these discs to a CR manifold with smooth dependence on parameters. By these tools we get extension of CR functions from wedges endowed with the sector property.     

CR Manifolds Admitting a CR Contraction

We classify the germs of C^￥\mathcal{C}^{\infty} CR manifolds that admit a smooth CR contraction. We show that such a CR manifold is embedded into ℂ ⁿ as a real hypersurface defined by a polynomial defining function consisting of monomials whose degrees are completely determined by the extended resonances of the contraction. Furthermore, the contraction map extends to a holomorphic contraction that coincides with its polynomial normal form. Consequently, several results concerning complex and CR geometry are derived.     

La dérivée schwarzienne en dynamique unimodale

The first return map of a C³ unimodal map to a neighbourhood of the critical point is conjugated by a real-analytic diffeomorphism to a map with negative Schwarzian derivative. Consequently, we obtain the classification of the metric attractors for smooth unimodal maps with non-degenerate critical point.     

Remarks on the generic rank of a CR mapping

We study germs of smooth CR mappings between embedded real hypersurfaces in complex spaces of the same dimension. In particular, we are interested in the generic rank of such mappings. IfH:M →M′ is a CR map between two hypersurfacesM andM′, we prove that ifM′ does not contain any germ of a holomorphic manifold then eitherH is constant or the generic rank ofH is odd. We also prove that if there is no formal holomorphic vector field tangent toM, then eitherH is constant or genericallyH is a local diffeomorphism. It follows, as a special case, that ifM andM′ are of D-finite type (in the sense of D’Angelo) thenH is either constant or is generically a local diffeomorphism. Supported by NSF Grant DMS 8901268.     

Local analysis near a folded saddle-node singularity

Folded saddle-nodes occur generically in one parameter families of singularly perturbed systems with two slow variables. We show that these folded singularities are the organizing centers for two main delay phenomena in singular perturbation problems: canards and delayed Hopf bifurcations. We combine techniques from geometric singular perturbation theory—the blow-up technique—and from delayed Hopf bifurcation theory—complex time path analysis—to analyze the flow near such folded saddle-nodes. In particular, we show the existence of canards as intersections of stable and unstable slow manifolds. To derive these canard results, we extend the singularly perturbed vector field into the complex domain and study it along elliptic paths. This enables us to extend the invariant slow manifolds beyond points where normal hyperbolicity is lost. Furthermore, we define a way-in/way-out function describing the maximal delay expected for generic solutions passing through a folded saddle-node singularity. Branch points associated with the change from a complex to a real eigenvalue structure in the variational equation along the critical (slow) manifold make our analysis significantly different from the classical delayed Hopf bifurcation analysis where these eigenvalues are complex only.     

Spherical space forms,CR mappings,and proper maps between balls

We determine precisely for which spherical space forms there are nontrivial smooth CR mappings to spheres. Equivalently we determine for which fixed point free finite unitary groups ? there exists a ?-invariant proper holomorphic rational map between balls. The answer is that the group must be cyclic and essentially only two classes of representations can occur. For these there are invariant polynomial examples.     

The Linearized Calderón Problem on Complex Manifolds

In this note we show that on any compact subdomain of a Kähler manifold that admits sufficiently many global holomorphic functions, the products of harmonic functions form a complete set. This gives a positive answer to the linearized anisotropic Calderón problem on a class of complex manifolds that includes compact subdomains of Stein manifolds and sufficiently small subdomains of Kähler manifolds. Some of these manifolds do not admit limiting Carleman weights, and thus cannot be treated by standard methods for the Calderón problem in higher dimensions. The argument is based on constructing Morse holomorphic functions with approximately prescribed critical points. This extends earlier results from the case of Riemann surfaces to higher dimensional complex manifolds.     

Analytic invariant manifolds for sequences of diffeomorphisms

We obtain real analytic invariant manifolds for trajectories of maps assuming only the existence of a nonuniform exponential behavior. We also consider the more general case of sequences of maps, which corresponds to a nonautonomous dynamics with discrete time. We emphasize that the maps that we consider are defined in a real Euclidean space, and thus, one is not able to obtain the invariant manifolds from a corresponding procedure to that in the nonuniform hyperbolicity theory in the context of holomorphic dynamics. We establish the existence both of stable (and unstable) manifolds and of center manifolds. As a byproduct of our approach we obtain an exponential control not only for the trajectories on the invariant manifolds, but also for all their derivatives.