Levi-flat Hypersurfaces Research Articles

Levi-flat hypersurfaces and their complement in complex surfaces

In this work we study analytic Levi-flat hypersurfaces in complex algebraic surfaces. First, we show that if this foliation admits chaotic dynamics (i.e. if it does not admit a transverse invariant measure), then the connected components of the complement of the hypersurface are modifications of Stein domains. This allows us to extend the CR foliation to a singular algebraic foliation on the ambient complex surface. We apply this result to prove, by contradiction, that analytic Levi-flat hypersurfaces admitting a transverse affine structure in a complex algebraic surface have a transverse invariant measure. This leads us to conjecture that Levi-flat hypersurfaces in complex algebraic surfaces that are diffeomorphic to a hyperbolic torus bundle over the circle are fibrations by algebraic curves.

LEVI-UMBILICAL REAL HYPERSURFACES IN A COMPLEX SPACE FORM

We give a classification of Levi-umbilical real hypersurfaces in a complex space form $\widetilde{M}_{n}(c)$, $n\geqslant 3$, whose Levi form is proportional to the induced metric by a nonzero constant. In a complex projective plane $\mathbb{C}\mathbb{P}^{2}$, we give a local construction of such hypersurfaces and moreover, we give new examples of Levi-flat real hypersurfaces in $\mathbb{C}\mathbb{P}^{2}$.

Compact Invariant Sets for Global Holomorhic Foliations

We study compact invariant sets for holomorphic foliations on Stein manifold. As application, we show some dynamical properties concerning minimal sets (with singularities) of foliations and real analytic Levi-flat hypersurfaces in projective spaces.

$1$-complete semiholomorphic foliations

A semiholomorphic foliations of type (n, d) is a differentiable real manifold X of dimension 2n + d, foliated by complex leaves of complex dimension n. In the present work, we introduce an appropriate notion of pseudoconvexity (and consequently, q-completeness) for such spaces, given by the interplay of the usual pseudoconvexity, along the leaves, and the positivity of the transversal bundle. For 1-complete real analytic semiholomorphic foliations, we obtain a vanishing theorem for the CR cohomology, which we use to show an extension result for CR functions on Levi flat hypersurfaces and an embedding theorem in C^N . In the compact case, we introduce a notion of weak positivity for the transversal bundle, which allows us to construct a real analytic embedding in CP^N .

Flattening of CR singular points and analyticity of the local hull of holomorphy I

This is the first article of the two papers in which we investigate the holomorphic and formal flattening problem for a codimension two real submanifold in $${\mathbb C}^n$$ with $$n\ge 3$$ near a non-degenerate CR singular point. The problem is motivated from the study of the complex Plateau problem that seeks for the Levi-flat hypersurface bounded by a given real submanifold and is motivated by the classical complex analysis problem of finding the local hull of holomorphy of a real submanifold in a complex space. The present article is focused on the case of CR singular points with at least one elliptic direction. We solve the holomorphic flattening problem and thus provide a complete description of the local hull of holomorphy in this setting. The results in this paper and those in (Flattening of CR singular points and analyticity of the local hull of holomorphy II, p. 60, 2014) are taken from our arxiv post (Flattening of CR singular points and analyticity of the local hull of holomorphy, 2012). We split (Flattening of CR singular points and analyticity of the local hull of holomorphy, 2012) into two independent articles to avoid it being too long.

Topology and dynamics of laminations in surfaces of general type

We study Riemann surface laminations in complex algebraic surfaces of general type. We focus on the topology and dynamics of minimal sets of holomorphic foliations and on Levi-flat hypersurfaces. We begin by providing various examples. Then our first result shows that Anosov Levi-flat hypersurfaces do not embed in surfaces of general type. This allows one to classify the possible Thurston’s geometries carried by Levi-flat hypersurfaces in surfaces of general type. Our second result establishes that minimal sets in surfaces of general type have a large Hausdorff dimension as soon as there exists a simply connected leaf. For both results, our methods rely on ergodic theory: we use harmonic measures and Lyapunov exponents.

Germs of singular Levi-flat hypersurfaces and holomorphic foliations

It is shown that the Levi foliation of a real analytic Levi-flat hypersurface extends to a d -web near a nondicritical singular point and admits a multiple-valued meromorphic first integral.

Singular Levi-flat hypersurfaces in complex projective space induced by curves in the Grassmannian

Let H ⊂ ℙn be a real-analytic subvariety of codimension one induced by a real-analytic curve in the Grassmannian G(n + 1, n). Assuming H has a global defining function, we prove H is Levi-flat, the closure of its smooth points of top dimension is a union of complex hyperplanes, and its singular set is either of dimension 2n - 2 or dimension 2n - 4. If the singular set is of dimension 2n - 4, then we show the hypersurface is algebraic and the Levi-foliation extends to a singular holomorphic foliation of ℙn with a meromorphic (rational of degree 1) first integral. In this case, H is in some sense simply a complex cone over an algebraic curve in ℙ1. Similarly if H has a degenerate singularity, then H is also algebraic. If the dimension of the singular set is 2n - 2 and is nondegenerate, we show by construction that the hypersurface need not be algebraic nor semialgebraic. We construct a Levi-flat real-analytic subvariety in ℙ2 of real codimension 1 with compact leaves that is not contained in any proper real-algebraic subvariety of ℙ2. Therefore a straightforward analogue of Chow's theorem for Levi-flat hypersurfaces does not hold.

Deformations of Levi flat hypersurfaces in complex manifolds

We first give a deformation theory of integrable distributions of codimension 1. We define a parametrization of families of smooth hypersurfaces near a Levi flat hypersurface L such that the Levi flat deformations are given by the solutions of the Maurer-Cartan equation in a DGLA associated to the Levi foliation. We say that L is infinitesimally rigid if the tangent cone at the origin to the moduli space of Levi flat deformations of L is trivial. We prove the infinitesimal rigidity of compact transversally parallelisable Levi flat hypersurfaces in compact complex manifolds and give sufficient conditions for infinitesimal rigidity in Kahler manifolds. As an application, we prove the nonexistence of transversally parallelizable Levi flat hypersurfaces in a class of manifolds which contains the complex projective plane.

Normal forms of Levi-flat hypersurfaces with Arnold type singularities

In this paper we study normal forms of Levi-flat hypersurfaces with singularities. We prove a result analogous to the Burns-Gong theorem for the existence of rigid normal forms of Levi-flat hypersurfaces which are defined by the vanishing of the real part of Ak , Dk , Ek singularities

On normal forms for Levi-flat hypersurfaces with an isolated line singularity

We prove the existence of normal forms for some local real-analytic Levi-flat hypersurfaces with an isolated line singularity. We also give sufficient conditions for that a Levi-flat hypersurface with a complex line as singularity to be a pullback of a real-analytic curve in C via a holomorphic function.

Classification of real analytic Levi flat hypersurfaces of 1-concave type in Hopf surfaces

A compact Levi flat hypersurface in a complex manifold is said to be of q-concave type if it admits a neighborhood system consisting of q-concave manifolds in the sense of Andreotti and Grauert. The real analytic Levi flat hypersurfaces of 1-concave type in Hopf surfaces are classified.

Boundary value problems and J-complex curves

We give a solution to the problem of filling by a Levi-flat hypersurface for a class of totally real tori in ℂ2 equipped with a certain almost complex structure.

Nonexistence of Certain Levi Flat Hypersurfaces in Kähler Manifolds from the Viewpoint of Positive Normal Bundles

Combining the $L^2$ method of solving the $\bar{\partial}$-equation with a method of Brunella in the study of Levi at hypersurfaces as stable sets of holomorphic foliations, two nonexistence theorems are proved for Levi flat hypersurfaces extending a result of Brunella.

Levi-Flat Hypersurfaces Tangent to Projective Foliations

Let M⊂ℙn be a singular real-analytic Levi-flat hypersurface tangent to a codimension-one holomorphic foliation \(\mathcal{F}\) on ℙn. 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.

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.

Boundaries of Levi-flat hypersurfaces: special hyperbolic points

Let $S\subset \C^n$, $n\geq 3$ be a compact connected 2-codimensional submanifold having the following property: there exists a Levi-flat hypersurface whose boundary is $S$, possibly as a current. Our goal is to get examples of such $S$ containing at least one special 1-hyperbolic point: sphere with two horns; elementary models and their gluing. The particular cases of graphs are also described.

Hartogs figure and symplectic non-squeezing

We solve a problem on filling by Levi-flat hypersurfaces for a class of totally real 2-tori in a real 4-manifold with an almost complex structure tamed by an exact symplectic form. As an application, we obtain a simple proof of Gromov’s non-squeezing theorem in dimension 4 and new results on rigidity of symplectic structures.

The Hartogs phenomenon on weakly pseudoconcave hypersurfaces

In 2002, Henkin and Michel proved a local Hartogs phenomenon for real analytic CR functions on real analytic weakly pseudoconcave CR manifolds. The aim of the present article is to remove the assumptions on real analyticity in the case of weakly pseudoconcave hypersurfaces \({M\subset\mathbb{C}^n}\) . If M is a graph of class \({\mathcal{C}^2}\) and n ≥ 3, a global theorem is proved for the extension of holomorphic germs along M. If the appearing domains have nicely shaped boundary, a Hartogs theorem even holds for continuous CR functions, where the difference to the case of holomorphic germs relies on the possible presence of lower-dimensional CR orbits. Levi flat hypersurfaces in \({\mathbb{C}^2}\) require a separate treatment. Here an affirmative answer is given to the question of Tomassini, whether 2-spheres bound 3-balls in M.