Related papers: On the Levi-flat Plateau problem

We study nonlinear automorphisms of Levi degenerate hypersurfaces of finite multitype. By recent results of Kolar, Meylan and Zaitsev, the Lie algebra of infinitesimal CR automorphisms may contain a graded component consisting of nonlinear…

Holomorphically homogeneous CR real hypersurfaces $M^3 \subset \mathbb{C}^2$ were classified by \'Elie Cartan in 1932. In the next dimension, we complete the classification of simply-transitive Levi non-degenerate hypersurfaces $M^5 \subset…

We show that every point in a uniformly $2$-nondegenerate CR hypersurface is canonically associated with a model $2$-nondegenerate structure. The $2$-nondegenerate models are basic CR invariants playing the same fundamental role as quadrics…

We discuss domains of holomorphy and several notions of pseudoconvexity (drawing parallels with the corresponding notions from geometric convexity), and present a mostly self-contained solution to the Levi problem. We restrict our attention…

We prove that if two real-analytic hypersurfaces in $\mathbb C^2$ are equivalent formally, then they are also $C^\infty$ CR-equivalent at the respective point. As a corollary, we prove that all formal equivalences between real-algebraic…

We classify locally defined non-spherical real-analytic hypersurfaces in complex space whose Levi form has no more than one negative eigenvalue and for which the dimension of the group of local CR-automorphisms has the second largest value.

We say that a CR singular submanifold $M$ has a removable CR singularity if the CR structure at the CR points of $M$ extends through the singularity as an abstract CR structure on $M$. We study such real-analytic submanifolds, in which case…

In this paper we shall give sufficient conditions for local CR diffeomorphisms between two real analytic submanifolds of $\Bbb C^N$ to be determined by finitely many derivatives at finitely many points. These conditions will also be shown…

Motivated by a question of Rubel, we consider the problem of characterizing which noncompact hypersurfaces in $\RR^n$ can be regular level sets of a harmonic function modulo a $C^\infty$ diffeomorphism, as well as certain generalizations to…

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…

We study normal forms of germs of singular real-analytic Levi-flat hypersurfaces. We prove the existence of rigid normal forms for singular Levi-flat hypersurfaces which are defined by the vanishing of the real part of complex…

We study singular real analytic Levi-flat subsets invariant by singular holomorphic foliations in complex projective spaces. We give sufficient conditions for a real analytic Levi-flat subset to be the pull-back of a semianalytic Levi-flat…

The ``complex Plateau problem'' (or boundary problem) in a complexe manifold X is the problem of characterizing the real submanifolds $\Gamma$ of X which are boundaries of analytic sub-varieties of $X \backslash \Gamma$. Our principal…

Hypersurface type CR-structures with non-degenerate Levi form on a manifold of dimension $(2n+1)$ have maximal symmetry dimension $n^2+4n+3$. We prove that the next (submaximal) possible dimension for a (local) symmetry algebra is $n^2+4$…

We reduce CR-structures on smooth elliptic and hyperbolic manifolds of CR-codimension 2 to parallelisms thus solving the problem of global equivalence for such manifolds. The parallelism that we construct is defined on a sequence of two…

We classify the Lie algebras of infinitesimal CR automorphisms of weakly pseudoconvex hypersurfaces of finite multitype in $\mathbb C^N$. In particular, we prove that such manifolds admit neither nonlinear rigid automorphisms, nor real or…

We extend the notion of a fundamental negatively $\mathbb Z$-graded Lie algebra $\mathfrak{m}_x=\bigoplus_{p\leq -1}\mathfrak{m}_x^p$ associated to any point of a Levi nondegenerate CR manifold to the class of $k$-nondegenerate CR manifolds…

Let $K = R$ or $C$. We study basic invariants of submanifolds of solutions $\mathcal{M} = \{ y = Q(x,a,b)\} = \{b = P(a,x,y)\}$ in coordinates $x \in K^{n\geqslant 1}$, $y \in K$, $a \in K^{m\geqslant 1}$, $b \in K$ under…

In this article we provide a version of Chow's theorem for real analytic Levi-flat hypersurfaces in the complex projective space $\mathbb{P}^{n}$, $n \geq 2$. More specifically, we prove that a real analytic Levi-flat hypersurface $M…

The purpose of this paper is to generalize in a geometric setting theorems of Severi, Brown and Bochner about analytic continuation of real analytic functions which are holomorphic or harmonic with respect to one of its variables. We prove…