Affine homogeneity of indefinite real hypersurfaces in the space ℂ3

We develop a constructive approach to the problem of describing affinely homogeneous real hypersurfaces in 3-dimensional complex space having nondegenerate sign-indefinite Levi form. We construct the affine invariants of a nondegenerate indefinite hypersurface in terms of second-order jets of its defining function and introduce the notion of the affine canonical equation of this surface. Three main types of canonical equations are considered. For each of these types, we construct a family of Lie algebras related to affinely homogeneous surfaces of a particular type. As a result, a family (depending on two real parameters) of affinely different homogeneous submanifolds of 3-dimensional complex space is presented (as matrix algebras).     

On holomorphic homogeneity of real hypersurfaces of general position in ℂ<Superscript>3</Superscript>

We study the problem of optimal recovery of a function analytic in a doubly connected domain from its approximately given values on one of the two components of the boundary. An optimal recovery method is obtained in the case when the error is an integer power of the modulus of the doubly connected domain.     

Contact metric hypersurfaces of ℂ2

Kaehler 2-form of ² being exact, it gives rise to a 1-form defined on a hypersurface of ². We show that if this 1-form defines a contact metric structure with respect to the induced metric on the compact and connected hypersurface of ², then the hypersurface is a round sphere of radius 1.This work is supported by Research Grant No. (Math/1409/05) of the Research Center, College of Science, King Saud University, Riyadh.     

Automorphisms of 2-Nondegenerate Hypersurfaces in ℂ3

An exact upper estimate is obtained for the dimension of the automorphism group of a 2-dimensional hypersurface in ³ possessing a Lie group structure.     

One-parameter families of operators in ℂ

We introduce classes of one-parameter families (OPF) of operators on C _c ^t8 (ℂ) which characterize the behavior of operators associated to the problem in the weighted space L² (ℂ, e^−2p) where p is a subharmonic, nonharmonic polynomial. We prove that an order 0 OPF operator extends to a bounded operator from L^q (ℂ) to itself, 1 < q < ∞, with a bound that depends on q and the degree of p but not on the parameter τ or the coefficients of p. Last, we show that there is a one-to-one correspondence given by the partial Fourier transform in τ between OPF operators of order m ≤ 2 and nonisotropic smoothing (NIS) operators of order m ≤ 2 on polynomial models in ℂ².     

Affine and Holomorphic Equivalence of Tube Domains in ℂ2

It is shown that, except for several explicitly described cases, two hyperbolic tube domains in $\mathbb{C}^2 $ are biholomorphically equivalent if and only if they are affinely equivalent.     

A characterization of totally <Emphasis Type=Italic>η</Emphasis>-umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form

We give a characterization of totally η-umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form in terms of totally umbilical condition for the holomorphic distribution on real hypersurfaces. We prove that if the shape operator A of a real hypersurface M of a complex space form M ⁿ (c), c ≠ 0, n ⩾ 3, satisfies g(AX, Y) = ag(X, Y) for any X, Y ∈ T ₀(x), a being a function, where T ₀ is the holomorphic distribution on M, then M is a totally η-umbilical real hypersurface or locally congruent to a ruled real hypersurface. This condition for the shape operator is a generalization of the notion of η-umbilical real hypersurfaces.     

The degrees of polynomial self-mappings of ℂ2

Two results on the degrees of polynomial mappings ²² are obtained.Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 527–534, April, 1998.     

Proper holomorphic self-maps of smooth bounded Reinhardt domains in ℂ2

It is proved that every proper holomorphic self-map of a smooth bounded Reinhardt domain in ?² is an automorphism.     

Boundary regularity of correspondences in ℂn

LetM, M′ be smooth, real analytic hypersurfaces of finite type in ℂⁿ and a holomorphic correspondence (not necessarily proper) that is defined on one side ofM, extends continuously up toM and mapsM to M′. It is shown that must extend acrossM as a locally proper holomorphic correspondence. This is a version for correspondences of the Diederich-Pinchuk extension result for CR maps.     

An Andreotti–Vesentini separation theorem on real hypersurfaces

Using Serre duality in CR manifolds and integral operators for the solution of the tangential Cauchy–Riemann equation with compact support, we prove a separation theorem of Andreotti–Vesentini type for the -cohomology in q-concave real hypersurfaces. Received: February 17, 1999?Published online: May 10, 2001     

On the\bar \partial equation in weightedL 2 norms in ℂ1equation in weightedL 2 norms in ℂ1

For a large class of subharmonicφ, the equation is studied in . Pointwise upper bounds are derived for the distribution kernels of the canonical solution operator and of the orthogonal projection onto the space of entire functions inH. Existence theorems inL ^p norms are derived as a corollary. A class of counterexamples, related to the failure of to be analytic-hypoelliptic on certain CR manifolds, is discussed. Communicated by Steven Krantz     

Instanton invariants of ℂP 2 via topology

Parts of this work were carried out while the first-named author was an EPSRC Visiting Fellow at The Isaac Newton Institute for Mathematical Sciences in Cambridge, and while the second-named author visited the University of Basel with support from the Swiss National Fund.     

Bichromatic and Equichromatic Lines in ℂ2 and ℝ2

Let G and R each be a finite set of green and red points, respectively, such that |G|=n, |R|=n−k, G∩R=∅, and the points of G∪R are not all collinear. Let t be the total number of lines determined by G∪R. The number of equichromatic lines (a subset of bichromatic) is at least (t+2n+3−k(k+1))/4. A slightly weaker lower bound exists for bichromatic lines determined by points in ℂ². For sufficiently large point sets, a proof of a conjecture by Kleitman and Pinchasi is provided. A lower bound of (2t+14n−k(3k+7))/14 is demonstrated for bichromatic lines passing through at most six points. Lower bounds are also established for equichromatic lines passing through at most four, five, or six points.