Related papers: Defining equations of $7$-dimensional model CR hyp…

We consider an alternative approach to a fundamental CR invariant - the Catlin multitype. It is applied to a general smooth hypersurface in $\mathbbC^{n+1}$, not necessarily pseudoconvex. Using this approach, we prove biholomorphic…

In this paper, motivated by the work of Kim and Kolar for the case of pseudoconvex models which are sums of squares of polynomials, we study the Lie algebra of real-analytic infinitesimal $CR$ automorphisms of a model hypersurface $M_0$…

We give a new proof of Faran's and Lebl's results by means of a new CR-geometric approach and classify all holomorphic mappings from the sphere in $\mathbb C^2$ to Levi-nondegenerate hyperquadrics in $\mathbb C^3$. We use the tools…

Let M be a connected real-analytic hypersurface in two dimensional complex space, $\mathbb C^2$, containing a connected complex hypersurface E, and let f be a smooth CR mapping sending M into another real-analytic hypersurface M' in…

Local conditions on boundaries of $C^\infty$ Levi-flat hypersurfaces, in case the boundary is a generic submanifold, are studied. For nontrivial real analytic boundaries we get an extension and uniqueness result, which forces the…

A curve is called nondegenerate if it can be modeled by a Laurent polynomial that is nondegenerate with respect to its Newton polytope. We show that up to genus 4, every curve is nondegenerate. We also prove that the locus of nondegenerate…

We show that the maximal prolongation of a certain algebra associated with a non-degenerate Hermitian form on ${\Bbb C}^n\times{\Bbb C}^n$ with values in ${\Bbb R}^k$ is canonically isomorphic to the Lie algebra of infinitesimal holomorphic…

We study a germ of real analytic n-dimensional submanifold of $C^n$ that has a complex tangent space of maximal dimension at a CR singularity. Under the condition that its complexification admits the maximum number of deck transformations,…

We study the deformation theory of CR maps in the positive codimensional case. In particular, we study structural properties of the {\em mapping locus} $E$ of (germs of nondegenerate) holomorphic maps $H \colon (M,p) \to M'$ between generic…

An explicit classification of simply connected compact homogeneous CR manifolds G/L of codimension one, with non-degenerate Levi form, is given. There are three classes of such manifolds: a) the standard CR homogeneous manifolds which are…

Let M be a connected real-analytic hypersurface in N-dimensional complex euclidean space whose Levi form is nondegenerate at some point. We prove that for every point p in M, there exists an integer k=k(M,p) such that germs at p of local…

We classify the tube domains in C^4 with affinely homogeneous base whose boundary contains a non-degenerate affinely homogeneous hypersurface. It follows that these domains are holomorphically homogeneous and amongst them there are four new…

We study curvature restrictions of Levi-flat real hypersurfaces in complex projective planes, whose existence is in question. We focus on its totally real Ricci curvature, the Ricci curvature of the real hypersurface in the direction of the…

In this paper, we obtain local smoothing estimates for the averages over nondegenerate surfaces of codimension $2$ in $\mathbb R^4$. We make use of multilinear restriction estimates and decoupling inequalities for a hypersurface in $\mathbb…

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 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…

Let $M \subset {\mathbb{C}}^{n+1}$, $n \geq 2$, be a real codimension two CR singular real-analytic submanifold that is nondegenerate and holomorphically flat. We prove that every real-analytic function on $M$ that is CR outside the CR…

Consider a $2$-nondegenerate constant Levi rank $1$ rigid $\mathcal{C}^\omega$ hypersurface $M^5 \subset \mathbb{C}^3$ in coordinates $(z, \zeta, w = u + iv)$: \[ u = F\big(z,\zeta,\bar{z},\bar{\zeta}\big). \] The Gaussier-Merker model…

We introduce a new class of quasi-linear parabolic equations involving nonhomogeneous degeneracy or/and singularity $$ \partial_t u=[|D u|^q+a(x,t)|D u|^s]\left(\Delta u+(p-2)\left\langle D^2 u\frac{D u}{|D u|},\frac{D u}{|D…

Let $Q^N_l\subset \bC\bP^{N+1}$ denote the standard real, nondegenerate hyperquadric of signature $l$ and $M\subset \bC^{n+1}$ a real, Levi nondegenerate hypersurface of the same signature $l$. We shall assume that there is a holomorphic…