Real Submanifolds In Complex Space Research Articles

On Beloshapka’s rigidity conjecture for real submanifolds in complex space

A well known Conjecture due to Beloshapka asserts that all totally nondegenerate polynomial models with the length $l \geq 3$ of their Levi–Tanaka algebra are rigid, that is, any point preserving automorphism of them is completely determined by the restriction of its differential at the fixed point onto the complex tangent space. For the length $l = 3$, Beloshapka’s Conjecture was proved by Gammel and Kossovskiy in 2006. In this paper, we prove the Conjecture for arbitrary length $l \geq 3$. As another application of our method, we construct polynomial models of length $l \geq 3$, which are not totally nondegenerate and admit large groups of point preserving nonlinear automorphisms.

On equivalence problem for 2–nondegenerate CR geometries with simple models

In this article, we solve the equivalence problem for 2–nondegenerate CR geometries that have (at every point) a homogeneous space G/H as a maximally symmetric model for G simple real Lie group of CR automorphisms. This completes the classification of real submanifolds in complex space that are maximally symmetric models with a real simple CR automorphism group. In particular, we construct (local) embeddings of these models into complex space.

Holomorphic versus algebraic equivalence for deformations of real-algebraic Cauchy–Riemann manifolds

Since Poincare’s celebrated paper [19] published in 1907, there has been a growing literature concerned with the equivalence problem for real submanifolds in complex space (see e.g., [4, 6, 7, 11, 13, 14, 22] for some recent works as well as the references therein). One interesting phenomenon, observed by Webster for biholomorphisms of Levi nondegenerate hypersurfaces [23], is that the biholomorphic equivalence of some types of real-algebraic submanifolds of a complex space implies their algebraic equivalence. In this paper, we show that this very phenomenon holds for algebraic deformations of germs of minimal holomorphically nondegenerate realalgebraic CR submanifolds in complex space. Let us recall that a germ of a real-algebraic CR submanifold (M,p) ⊂ (Cn, p) is minimal if there exists no proper CR submanifold N ⊂ M through p of the same CR dimension as M . It is holomorphically nondegenerate if there exists no nontrivial holomorphic vector field tangent to M near p (see [21]). An algebraic deformation of (M,p) is a real-algebraic family of germs at p of real-algebraic CR submanifolds (Ms, p)s∈Rk in Cn, defined for s ∈ Rk near 0, such that M0 = M . We say that two such deformations (Ms, p)s∈Rk and (Nt, p′)t∈Rk are biholomorphically equivalent if there exists a germ of a realanalytic diffeomorphism φ : (Rk, 0) → (Rk, 0) and a holomorphic submersion B : (Cz × Cu, (p, 0)) → (Cn, p′) such that z → B(z, s) is a biholomorphism sending (Ms, p) to (Nφ(s), p′) for all s ∈ Rk close to 0. We shall say that such

Geometric Inequalities for Purely Real Submanifolds in Complex Space Forms

As a generalisation of Kaehlerian slant submanifolds in Kaehler manifolds (i.e., proper slant submanifolds with the canonical endomorphism P parallel, ∇P = 0) one considers purely real submanifolds with ∇P = 0. The class of purely real submanifolds with ∇P = 0 also contains totally real submanifolds, in particular Lagrangian submanifolds. We obtain Chen-like inequalities for purely real submanifolds in complex space forms, i.e., relationships between intrinsic and extrinsic invariants of such submanifolds, involving the scalar curvature and Chen first invariant, respectively, and the squared mean curvature and the holomorphic sectional curvature of the ambient space.

Representation of the group of holomorphic symmetries of a real germ in the symmetry group of the model surface of the germ

There is a series of papers devoted to the construction and investigation of local polynomial models of real submanifolds of complex space (see, e.g., [1]). Among the main properties of model surfaces, we can mention the following fact. The dimension of the local group of holomorphic symmetries of a germ does not exceed the dimension of the similar group for the tangent model surface of the germ. In the present note, this assertion is presented in a much stronger form. A faithful representation of the Lie algebra of infinitesimal automorphisms of a germ in the Lie algebra of the tangent model surface is constructed.

Real congruence of complex matrix pencils and complex projections of real Veronese varieties

Quadratically parametrized maps from a real projective space to a complex projective space are constructed as projections of the Veronese embedding. A classification theorem relates equivalence classes of projections to real congruence classes of complex symmetric matrix pencils. The images of some low-dimensional cases include certain quartic curves in the Riemann sphere, models of the real projective plane in complex projective 4-space, and some normal form varieties for real submanifolds of complex space with CR singularities.

On some rigidity properties of mappings between CR-submanifolds in complex space

We survey some recent results on holomorphic or formal mappings sending real submanifolds in complex space into each other. More specifically, the approximation and convergence properties of formal CR-mappings between real-analytic CR-submanifolds will be discussed, as well as the corresponding questions in the category of real-algebraic CR-submanifolds.

Equivalences of Real Submanifolds in Complex Space

We show that for any real-analytic submanifold M in ℂN there is a proper real-analytic subvariety V ⊂ M such that for any p ∊ M \ V, any realanalytic submanifold M′ in ℂN, and any p′ ∊ M′, the germs (M, p) and (M′, p′) of the submanifolds M and M′ at p and p′ respectively are formally equivalent if and only if they are biholomorphically equivalent. As an application, for p ∊ M \ V, the problem of biholomorphic equivalence of the germs (M, p) and (M′, p′) is reduced to that of solving a system of polynomial equations. More general results for k-equivalences are also stated and proved.

Polynomial models of real manifolds

We construct polynomial models for germs of real submanifolds in complex space. It was shown earlier that the properties of models of degree 3 (for appropriate values of the codimension) are similar to well-known properties of tangent quadrics. In this paper we construct models of arbitrarily high degree. They have all these properties with one exception: from degree 5 onwards, they are not completely universal.

Book Review: Real submanifolds in complex space and their mappings

Book Review: Real submanifolds in complex space and their mappings

Local geometric properties of real submanifolds in complex space

We survey some recent results on local geometric properties of real submanifolds of complex space. Our main focus is on the structure and properties of mappings between such submanifolds. We relate these results to the classification of real submanifolds under biholomorphic, algebraic, or formal transformations. Examples and open problems in this context are also mentioned.

On removable singularities for integrable cr functions

Let $M$ be a locally embeddable CR manifold. One defines, for $p\geq 1$, $$L^p_{\rm loc,CR}(M)=\{f\in L^p_{\rm loc}(M)| f {\rm is CR on} M\},$$ $$L^p_{\rm loc,CR}(M-E)=\{f\in L^p_{\rm loc}(M)| f {\rm is CR on} M-E\},$$ with $E$ a closed subset of $M$. This paper gives conditions under which $L^p_{\rm loc,CR}(M)=L^p_{\rm loc,CR}(M-E)$. In order to state results, let us recall the following definitions. Definition 1: The CR orbit of $p\in M$ is defined as $O_{\rm CR}(M,p)=\{q\in M|$ there exists a piecewise smooth integral curve of $T^cM$ with origin $p$ and target $q\}$. By Sussmann's theorem, $O_{\rm CR}(M,p)$ has the structure of a smooth immersed submanifold of $M$ [see M. S. Baouendi, P. Ebenfelt and L. P. Rothschild, Real submanifolds in complex space and their mappings, Princeton Univ. Press, Princeton, NJ, 1999; MR1668103 (2000b:32066)]. Definition 2: $M$ is globally minimal if $M$ is a single orbit. Let $H^k$ denote $k$-dimensional Hausdorff measure. The main general assumption throughout the paper is that $M$ and $M-E$ are globally minimal ($M$ globally minimal does not imply $M-E$ globally minimal). Among the results of the paper, we state the following theorem: Let $M$ be $C^{2,\alpha}, 0<\alpha<1$, of real dimension $d$ and CR dimension $\geq 1$. Let $E\subset M$ be a closed subset with $H^{d-3}(E)<\infty$. Assume that $M$ and $M-E$ are globally minimal. Then $L^p_{\rm loc,CR}(M)=L^p_{\rm loc,CR}(M-E)$.