Smooth Real Hypersurface Research Articles

On holomorphic foliations admitting invariant CR manifolds

We study holomorphic foliations of codimension $k\geq 1$ on a complex manifold $X$ of dimension $n+k$ from the point of view of the exceptional minimal set conjecture. For $n\geq 2$ we show in particular that if the holomorphic normal bundle $N_{\mathcal{F}}$ is Griffiths positive, then the foliation does not admit a compact invariant set that is a complete intersection of $k$ smooth real hypersurfaces in $X$.

Bit complexity for computing one point in each connected component of a smooth real algebraic set

We analyze the bit complexity of an algorithm for the computation of at least one point in each connected component of a smooth real algebraic set. This work is a continuation of our analysis of the hypersurface case (On the bit complexity of finding points in connected components of a smooth real hypersurface, ISSAC'20). In this paper, we extend the analysis to more general cases.Let F=(f1,…,fp) in Z[X1,…,Xn]p be a sequence of polynomials with V=V(F)⊂Cn a smooth and equidimensional variety and 〈F〉⊂C[X1,…,Xn] a radical ideal. To compute at least one point in each connected component of V∩Rn, our starting point is an algorithm by Safey El Din and Schost (Polar varieties and computation of one point in each connected component of a smooth real algebraic set, ISSAC'03). This algorithm uses random changes of variables that are proven to generically ensure certain desirable geometric properties. The cost of the algorithm was given in an algebraic complexity model; here, we analyze the bit complexity and the error probability, and we provide a quantitative analysis of the genericity statements. In particular, we are led to use Lagrange systems to describe polar varieties, as they make it simpler to rely on techniques such as weak transversality and an effective Nullstellensatz.

Nonlinear CR automorphisms of Levi degenerate hypersurfaces and a new gap phenomenon

We give a complete classification of polynomial models for smooth real hypersurfaces of finite Catlin multitype in C-3, which admit nonlinear infinitesimal CR automorphisms. As a consequence, we obtain a sharp 1-jet determination result for any smooth hypersurface with such a model. The results also prove a conjecture of the first author about the origin of such nonlinear automorphisms (AIM list of problems, 2010). As another consequence, we describe all possible dimensions of the Lie algebra of infinitesimal CR automorphisms, which leads to a new secondary gap phenomenon.

A Note on Infinite Type Germs of a Real Hypersurface in

Abstract: The purpose of this article is to show that there exists a smooth real hypersurface germ of D'Angelo infinite type in such that it does not admit any (singular) holomorphic curve that has infinite order contact with at .
 2010 Mathematics Subject Classification. Primary 32T25; Secondary 32C25.
 Key words and phrases: Holomorphic vector field, automorphism group, real hypersurface, infinite type point.

A geometric approach to Catlin’s boundary systems

For a point $p$ in a smooth real hypersurface $M\subset\C^n$, where the Levi form has the nontrivial kernel $K^{10}_p$, we introduce an invariant cubic tensor $\tau^3_p \colon \C T_p \times K^{10}_p \times \overline{K^{10}_p} \to \C\otimes (T_p/H_p)$, which together with Ebenfelt's tensor $\psi_3$, constitutes the full set of $3$rd order invariants of $M$ at $p$. Next, in addition, assume $M\subset\C^n$ to be {\em (weakly) pseudoconvex}. Then $\tau^3_p$ must identically vanish. In this case we further define an invariant quartic tensor $\tau^4_p \colon \C T_p \times \C T_p \times K^{10}_p\times \overline{K^{10}_p} \to \C\otimes (T_p/H_p)$, and for every $q=0, \ldots, n-1$, an invariant submodule sheaf of $(1,0)$ vector fields in terms of the Levi form, and an invariant ideal sheaf of complex functions generated by certain derivatives of the Levi form, such that the set of points of Levi rank $q$ is locally contained in certain real submanifolds defined by real parts of the functions in the ideal sheaf, whose tangent spaces have explicit algebraic description in terms of the quartic tensor $\tau^4$. Finally, we relate the introduced invariants with D'Angelo's finite type, Catlin's mutlitype and Catlin's boundary systems.

On the CR automorphism group of a certain hypersurface of infinite type in ℂ2

We consider -smooth real hypersurfaces of infinite type in . The purpose of this paper is to give explicit descriptions for stability groups of the hypersurface and a radially symmetric hypersurface in .

The Cartan equivalence problem for Levi-non-degenerate real hypersurfaces

We develop the Cartan equivalence problem for Levi-non- degenerate -smooth real hypersurfaces in in great detail, performing all computations effectively in terms of local graphing functions. In particular, we present explicitly the unique (complex) essential invariant of the problem. Comparison with our previous joint results [1] shows that the Cartan–Tanaka geometry of these real hypersurfaces perfectly matches their biholomorphic equivalence.

Common boundary values of holomorphic functions for two-sided complex structures

L et� 1, � 2 be two disjoint open sets in R 2n whose bound- aries share a smooth real hypersurface M as a relatively open sub- set. Assume thati is equipped with a complex structure J i that is smooth up to M. Suppose that at each point x ∈ M there is a vector v ∈ Tx M such that J 1 x v and J 2 x v are in the same connected component of TxR 2n Tx M.I ff is holomorphic with respect to both structures in the open sets and continuous on � 1 ∪ M ∪ � 2 ,t henf must be smooth on the union � 1 ∪ M. Although the result, as stated, is far more meaningful for integrable structures, our methods make it much more natural to deal with the general almost complex structures with- out the integrability condition. The result is therefore proved in the framework of almost complex structures.

Formal complex curves in real smooth hypersurfaces

In this paper, we investigate germs of smooth real hypersurfaces in $\mathbb{C}^{n}$. We show that if the hypersurface is of infinite D’Angelo type at a point, then there exists a formal complex curve in the hypersurface through that point.

SOBOLEV ESTIMATES FOR THE LOCAL EXTENSION OF BOUNDARY HOLOMORPHIC FORMS ON REAL HYPERSURFACES IN ℂn

Let M be a smooth real hypersurface in complex space of dimension <TEX>$n$</TEX>, <TEX>$n{\geq}3$</TEX>, and assume that the Levi-form at <TEX>$z_0$</TEX> on M has at least <TEX>$(q+1)$</TEX>-positive eigenvalues, <TEX>$1{\leq}q{\leq}n-2$</TEX>. We estimate solutions of the local <TEX>$\bar{\partial}$</TEX>-closed extension problem near <TEX>$z_0$</TEX> for <TEX>$(p,q)$</TEX>-forms in Sobolev spaces. Using this result, we estimate the local solution of tangential Cauchy-Riemann equation near <TEX>$z_0$</TEX> in Sobolev spaces.

Transversality of holomorphic mappings between real hypersurfaces in complex spaces of different dimensions

We consider holomorphic mappings $H$ between a smooth real hypersurface $M\subset \bC^{n+1}$ and another $M'\subset \bC^{N+1}$ with $N\geq n$. We provide conditions guaranteeing that $H$ is transversal to $M'$ along all of $M$. In the strictly pseudoconvex case, this is well known and follows from the classical Hopf boundary lemma. In the equidimensional case ($N=n$), transversality holds for maps of full generic rank provided that the source is of finite type in view of recent results by the authors (see also a previous paper by the first author and L. Rothschild). In the positive codimensional case ($N>n$), the situation is more delicate as examples readily show. In recent work by S. Baouendi, the first author, and L. Rothschild, conditions were given guaranteeing that the map $H$ is transversal outside a proper subvariety of $M$, and examples were given showing that transversality may fail at certain points. One of the results in this paper implies that if $N\le 2n-2$, $M'$ is Levi-nondegenerate, and $H$ has maximal rank outside a complex subvariety of codimension 2, then $H$ is transversal to $M'$ at all points of $M$. We show by examples that this conclusion fails in general if $N\geq 2n$, or if the set $W_H$ of points where $H$ is not of maximal rank has codimension one. We also show that $H$ is transversal at all points if $H$ is assumed to be a finite map (which allows $W_H$ to have codimension one) and the stronger inequality $N\leq 2n-3$ holds, provided that $M$ is of finite type.

Sobolev estimates for the local extension of ∂̄_{𝑏}-closed (0,1)-forms on real hypersurfaces in ℂⁿ with two positive eigenvalues

Let M \mathcal M be a smooth real hypersurface in complex space of dimension n ≥ 3 n\ge 3 , and assume that the Levi-form at z 0 z_0 on M \mathcal M has at least two positive eigenvalues. We estimate solutions of the local ∂ ¯ \bar {\partial } -closed extension problem near z 0 z_0 for ( 0 , 1 ) (0,1) -forms in Sobolev spaces. Using this result, we estimate the local solution of tangential Cauchy-Riemann equations near z 0 z_0 for ( 0 , 1 ) (0,1) -forms in Sobolev spaces.

CR functions on subanalytic hypersurfaces

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 particular, sufficient geometric conditions are given for a point p on a subanalytic hypersurface M to admit a germ at p of a smooth CR function f that cannot be holomorphically extended to either side of M. As a consequence it is shown that a well-known condition of the absence of complex hypersurfaces contained in a smooth real hypersurface M, which guarantees one-sided holomorphic extension of CR functions on M, is neither a necessary nor a sufficient condition for one-sided holomorphic extension in the singular case.

Finite jet determination of holomorphic mappings at the boundary

Let M,M' be smooth real hypersurfaces in N-dimensional space and assume that M is k-nondegenerate at a point p in M. We prove that holomorphic mappings that extend smoothly to M, sending a neighborhood of p in M diffeomorphically into M' are completely determined by their 2k-jet at p. As an application of this result, we also give sufficient conditions on a smooth real hypersurface which guarantee that the space of infinitesimal CR automorphisms is finite dimensional.

Smooth Real Hypersurfaces in ℂn with a Non-compact Isotropy Group of CR Transformations

Let M 2n+1 be an (abstract) smooth, real hypersurface of ℂ n+1 , strongly pseudoconvex, and let us assume that there exists a non-compact CR transformation f fixing a point p ∈ M 2n+1 . We prove that there exists an f-contractible neighborhoodU of p (i.e. the closure of any orbit f n (u) with u ∈ U contains p). This gives an elementary proof of the following recent result by R. Schoen: a strongly pseudoconvex smooth real hypersurface in ℂ 2n+1 , admitting a non-compact transformation fixing a point, is globally equivalent to S 2n+1 or to S 2n+1 with a point deleted.

Mappings of real algebraic hypersurfaces

A holomorphic function h(Z) defined in a neighborhood of a point Z0 E CN is algebraic if there are polynomials qj(Z), j = 0, ..., k, not all identically zero, such that qk(Z)h(Z)k + ... + qo(Z) _ 0. A holomorphic mapping is algebraic if all its components are algebraic. A smooth real hypersurface M in CN is algebraic if it is contained in the zero set of a nontrivial real-valued polynomial in Z and Z. We assume throughout this paper that N > 1. In [WI], Webster proved the following celebrated result: If M and M' are algebraic hypersurfaces in CN with nondegenerate Levi forms and if H is a biholomorphism defined in an open neighborhood of M and mapping M into M', then H is algebraic. Previously, it had long been known that if M and M are open subsets of spheres, then the components of H are in fact rational functions (Poincare [P], Tanaka [T]). In this paper we go a step further and give a complete characterization of algebraic hypersurfaces in CN such that any holomorphic mapping with nonvanishing Jacobian determinant between two such hypersurfaces must be algebraic. Our main result is stated in Theorem 1 below. ,N By a germ at p0 of a holomorphic vector field in C , we shall mean a

The notion of formal essential finiteness for smooth real hypersurfaces

The notion of formal essential finiteness for smooth real hypersurfaces

Local boundary regularity of holomorphic mappings

AbstractLet f be a holomorphic mapping between two bounded domains D and D' in complex space ℂn. Suppose that D and D' contain smooth real hypersurfaces Γ and Γ′ as open subsets of their respective boundaries, which correspond under a continuous extension of f. We shall show that this extension is smooth, given certain restrictions on Γ, Γ, and f.