Related papers: Boundary value problems and J-complex curves

We establish an inequality among the Ricci curvature, the squared mean curvature, and the normal curvature for real hypersurfaces in complex space forms. We classify real hypersurfaces in two-dimensional non-flat complex space forms which…

In this paper the Navier problem and the Dirichlet problem for Willmore curves in $\mathbb{R}^2$ is solved.

We study pseudoholomorphic discs with boundaries attached to a real hypersurface in an almost complex manifold. We give sufficient conditions for filling a one sided neighborhood of the hypersurface by the discs.

Normal forms of almost complex structures in a neighborhood of pseudoholomorphic curve are considered. We define normal bundles of such curves and study the properties of linear bundle almost complex structures. We describe 1-jet of the…

Let $H \subset {\mathbb P}^n$ 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…

We discuss local polynomial convexity of real analytic Levi-flat hypersurfaces in $\mathbb C^n$, $n>1$, near singular points.

The purpose of this article is to classify the real hypersurfaces in complex space forms of dimension 2 that are both Levi-flat and minimal. The main results are as follows: When the curvature of the complex space form is nonzero, there is…

We solve the $\partial \bar{\partial}$-problem for a form with distribution boundary value on a Levi flat unbounded domain of $\mathbb{C}^n$ with the complementary is also Levi flat and unbounded.

We introduce a class of combinatorial hypersurfaces in the complex projective space. They are submanifolds of codimension~2 in $\C P^n$ and are topologically "glued" out of algebraic hypersurfaces in $(\C^*)^n$. Our construction can be…

We establish a JSJ-type decomposition theorem for splitting exact symplectic fillings of contact 3-manifolds along \emph{mixed tori} -- these are convex tori satisfying a particular geometric condition. As an application, we show that if…

There are solved standard problems related to Formal (Holomorphic) Segre preserving Mappings of non-trivial Real-Formal Hypersurfaces in $\mathbb{C}^{2}$.

We investigate the problem of finding complete strictly convex hypersurfaces of constant curvature in hyperbolic space with a prescribed asymptotic boundary at infinity for a general class of curvature functions.

We investigate the logarithmic bundles associated to arrangements of hypersurfaces with a fixed degree in a smooth projective variety. We then specialize to the case when the variety is a quadric hypersurface and a multiprojective space to…

We classify real hypersurfaces in CP^2and CH^2 equipped with pseudo-parallel structure Jacobi operator.

We show the existence of a complex K3 surface $X$ which is not a Kummer surface and has a one-parameter family of Levi-flat hypersurfaces in which all the leaves are dense. We construct such $X$ by patching two open complex surfaces…

We continue our study, initiated in an earlier article, of a class of rigid hypersurfaces in ${\mathbb C}^3$ that are 2-nondegenerate and uniformly Levi degenerate of rank 1, having zero CR-curvature. We drop the restrictive assumptions of…

The complex Plateau problem is analogous, in a Hermitian complex manifold, to the classical Plateau problem in 3 dimensional real space: it is a geometrical problem of extension of a closed real manifold into a complex analytic subvariety,…

In \cite{DJL07} it was shown that if $\scra$ is an affine hyperplane arrangement in $\C^n$, then at most one of the $L^2$--Betti numbers $b_i^{(2)}(\C^n\sm \scra,\id)$ is non--zero. In this note we prove an analogous statement for…

We prove that each point in an almost-complex surface has a basis of complete hyperbolic neighborhoods. The problem is local, and therefore we can consider the case when our surface is ${\bf R^4}$ with an arbitrary almost-complex structure…

In this paper we consider germs of smooth Levi flat hypersurfaces, under the following notion of local equivalence: S_1 ~ S_2 if their one-sided neighborhoods admit a biholomorphism smooth up to the boundary. We introduce a simple invariant…