Related papers: On the Levi-flat Plateau problem
Plateau's problem is not a single conjecture or theorem, but rather an abstract framework, encompassing a number of different problems in several related areas of mathematics. In its most general form, Plateau's problem is to find an…
For certain real hypersurfaces in the projective space, of signature (1,n), we study the filling problem for small deformations of the CR structure (the other signatures being well understood). We characterize the deformations which are…
We construct complete normal forms for $5$-dimensional real hypersurfaces in $\mathbb C^3$ which are $2$-nondegenerate and also of Levi non-uniform rank zero at the origin point ${\bf p} =0$. The latter condition means that the rank of the…
We study the local equivalence problem for five dimensional real hypersurfaces $M^5$ of $\mathbb{C}^3$ which are $2$-nondegenerate and of constant Levi rank $1$ under biholomorphisms. We find two invariants, $J$ and $W$, which are expressed…
We study the Plateau problem with a lower dimensional obstacle in $\mathbb{R}^n$. Intuitively, in $\mathbb{R}^3$ this corresponds to a soap film (spanning a given contour) that is pushed from below by a "vertical" 2D half-space (or some…
A connected real analytic hypersurface M in C^(n+1) whose Levi form is nondegenerate in at least one point - hence at every point of some Zariski-open subset - is locally biholomorphic to the model Heisenberg quadric pseudosphere of…
This article concerns a natural generalization of the classical asymptotic Plateau problem in hyperbolic space. We prove the existence of a smooth complete hypersurface of constant scalar curvature with a prescribed asymptotic boundary at…
Given a complete Riemannian manifold $\mathcal{M}\subset\mathbb{R}^d$ which is a Lipschitz neighbourhood retract of dimension $m+n$, of class $C^{h,\beta}$ and an oriented, closed submanifold $\Gamma \subset \mathcal M$ of dimension $m-1$,…
Searching normal forms for real analytic submanifolds of C^n involves convergence problems. In 1983, J.K. Moser and S.M. Webster provided examples of real analytic surfaces in C^2 having an isolated hyperbolic (in the sense of E. Bishop)…
On the basis of a result of Barrett, we show that members of certain classes of abstract Levi flat manifolds with boundary, whose Levi foliation contains a compact leaf with contracting, flat holonomy, admit no $CR$ embedding as a…
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…
An example is given of a hyperconvex manifold without non-constant bounded holomorphic functions, which is realized as a domain with real-analytic Levi-flat boundary in a projective surface.
In continuing the study of harmonic mapping from 2-dimensional Riemannian simplicial complexes in order to construct minimal surfaces with singularity, we obtain an a-priori regularity result concerning the real analyticity of the free…
In this paper, we study the real hypersurfaces $M$ in $\mathbb C^2$ at points $p\in M$ of infinite type. The degeneracy of $M$ at $p$ is assumed to be the least possible, namely such that the Levi form vanishes to first order in the CR…
A general class of singular real hypersurfaces, called subanalytic, is defined. For a subanalytic hypersurface M in C^n, Cauchy-Riemann (or simply CR) functions on M are defined, and certain properties of CR functions discussed. In…
Let $M$ be a connected real-analytic hypersurface in $\C^N$ and $\S$ the unit real sphere in $\C^{N'}$, $N'> N\geq 2$. Assume that $M$ does not contain any complex-analytic hypersurface of $\C^N$ and that there exists at least one strongly…
Solving the Plateau problem means to find the surface with minimal area among all surfaces with a given boundary. Part of the problem actually consists of giving a suitable definition to the notions of 'surface', 'area' and 'boundary'. In…
CR singularities of real 4-submanifolds in complex 3-space are classified by using local holomorphic coordinate changes to transform the quadratic coefficients of the real analytic defining equation into a normal form. The quadratic…
Given a C2-domain with compact boundary in an arbitrary complete Riemannian manifold, we search for smallness conditions on the boundary data for which the Dirichlet problem for the minimal hypersurface equation is solvable. We obtain an…
We provide explicit counterexamples to the so-called Complement Problem in every dimension $n\geq3$, i.e. pairs of non-isomorphic irreducible hypersurfaces $H_1, H_2\subset\mathbb{C}^{n}$ whose complements $\mathbb{C}^{n}\setminus H_1$ and…