Related papers: The Poincare lemma and local embeddability

Let $M^{2n+1}$ ($n \geq 2$) be a compact pseudoconvex CR manifold of finite commutator type whose $\dbarb$ has closed range in $L^2$ and whose Levi form has comparable eigenvalues. We prove a sharp $L^1$ Sobolev inequality for the $\dbarb$…

In this paper we introduce the concept of inflexible $CR$ submanifolds. These are $CR$ submanifolds of some complex Euclidean space such that any compactly supported $CR$ deformation is again globally $CR$ embeddable into some complex…

We propose a new method for obtaining Poincare-type inequalities on arbitrary convex bodies in R^n. Our technique involves a dual version of Bochner's formula and a certain moment map, and it also applies to some non-convex sets. In…

Local CR-generic submanifolds of C^N are in one-to-one correspondence with their respective graphing functions, but it is well known that (despite their importance) the Cartan-Hachtroudi-Chern-Moser invariants and coframes for Levi…

The totally-real embeddability of any $2k$-dimensional compact manifold $M$ into $\mathbb C^n$, $n\geq 3k$, has several consequences: the genericity of polynomially convex embeddings of $M$ into $\mathbb C^n$, the existence of $n$ smooth…

In this paper, we consider real hypersurfaces $M$ in $\Bbb C^3$ (or more generally, 5-dimensional CR manifolds of hypersurface type) at uniformly Levi degenerate points, i.e. Levi degenerate points such that the rank of the Levi form is…

We study compactness and noncompactness phenomena for the CR Yamabe equation on compact strictly pseudoconvex CR manifolds. First, in dimension five we establish uniform \emph{a priori} estimates for families of positive solutions of…

We show that every closed L_infty,loc - form on R^n is exact. Differential is understood in the sense of currents. The proof does not use any explicit geometric constructions. De Rham theorem follows.

We prove a Poincar\'e, and a general Sobolev type inequalities for functions with compact support defined on a $k$-rectifiable varifold $V$ defined on a complete Riemannian manifold with positive injectivity radius and sectional curvature…

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…

Among its many corollaries, Poincare duality implies that the de Rham cohomology of a compact oriented manifold is a shifted commutative Frobenius algebra --- a commutative Frobenius algebra in which the comultiplication has cohomological…

We show that, for a closed orientable n-manifold, with n not congruent to 3 modulo 4, the existence of a CR-regular embedding into complex (n-1)-space ensures the existence of a totally real embedding into complex n-space. This implies that…

We show that for a real-analytic connected holomorphically nondegenerate 5-dimensional CR-hypersurface $M$ and its symmetry algebra $\mathfrak{s}$ one has either: (i) $\dim\mathfrak{s}=15$ and $M$ is spherical (with Levi form of signature…

The first part of this paper considers higher order CR invariants of three dimensional hypersurfaces of finite type. Using a full normal form we give a complete characterization of hypersurfaces with trivial local automorphism group, and…

Contrary to the expected behavior, we show the existence of non-invertible deformations of Lie algebras which can generate invariants for the coadjoint representation, as well as delete cohomology with values in the trivial or adjoint…

We present a differential geometric formulation of the Poincare problem using the calculus of moving surfaces (CMS). In this framework, an n dimensional compact hypersurface evolves under a velocity field that couples motion to the…

We classify polynomial models for real hypersurfaces in $\mathbb C^N$, which admit nonlinearizable infinitesimal CR automorphisms. As a consequence, this provides an optimal 1-jet determination result in the general case. Further we prove…

We show that the Poincar\'e lemma we proved elsewhere in the context of crystalline cohomology of higher level behaves well with regard to the Hodge filtration. This allows us to prove the Poincar\'e lemma for transversal crystals of level…

We define and study a class of finite topological spaces, which model the cell structure of a space obtained by gluing finitely many Euclidean convex polyhedral cells along congruent faces. We call these finite topological spaces,…

We show that any compact smooth real $n$-dimensional manifold $M$ with $n\leq 11$ can be smoothly embedded into $\mathbb{C}^{n+1}$ as a polynomially convex set. In general, there is no such embedding into $\mathbb{C}^n$. This solves a…