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Rasul Shafikov

Publications and source records attributed to Rasul Shafikov.

At least 37 records · Page 2Linked to original sources
arXiv2016

Rational and Polynomial Density on Compact Real Manifolds

We establish a characterization for an $m$-manifold $M$ to admit $n$ functions $f_1$,...,$f_n$ and $n'$ functions $g_1,...,g_{n'}$ in $\mathcal{C}^\infty(M)$ so that every element of $\mathcal{C}^k(M)$ can be approximated by rational combinations of $f_1,...,f_n$ and polynomial combinations of $g_1,...,g_{n'}$. As an application, we show that the optimal value of $n$ and $n'$ for all manifolds of dimension $m$ is [3m/2], when $k\geq 1$ and $m\geq 2$.

math.CV↗
arXiv2015

Distributional Boundary Values: Some New Perspectives

We discuss without proof some recent advances in the theory of distributional boundary values, and give an example to show that the condition that a piecewise smooth domain has generic corners is necessary for the existence of boundary values.

math.CV↗
arXiv2015

Distributional boundary values of holomorphic functions on product domains

We show that holomorphic functions of polynomial growth on domains with corners have distributional boundary values in an appropriate sense, provided the corners are generic CR manifolds. We prove an analog of the Bochner-Hartogs theorem for these boundary values for the simplest such domains, the product domains.

math.CV↗
arXiv2014

Critical sets of proper holomorphic mappings

It is shown that if a proper holomorphic map $f: \mathbb C^n \to \mathbb C^N$, $1<n\le N$, sends a pseudoconvex real analytic hypersurface of finite type into another such hypersurface, then any $n-1$ dimensional component of the critical locus of $f$ intersects both sides of $M$. We apply this result to the problem of boundary regularity of proper holomorphic mappings between bounded domains in $\mathbb C^n$.

math.CV↗
arXiv2014

Analytic Differential Equations and Spherical Real Hypersurfaces

We establish an injective correspondence $M\longrightarrow\mathcal E(M)$ between real-analytic nonminimal hypersurfaces $M\subset\mathbb{C}^{2}$, spherical at a generic point, and a class of second order complex ODEs with a meromorphic singularity. We apply this result to the proof of the bound $\mbox{dim}\,\mathfrak{hol}(M,p)\leq 5$ for the infinitesimal automorphism algebra of an \it arbitrary \rm germ $(M,p)\not\sim(S^3,p')$ of a real-analytic Levi nonflat hypersurface $M\subset\mathbb{C}^2$ (the Dimension Conjecture). This bound gives the first proof of the dimension gap $\mbox{dim}\,\mathfrak{hol}(M,p)=\{8,5,4,3,2,1,0\}$ for the dimension of the automorphism algebra of a real-analytic Levi nonflat hypersurface. As another application we obtain a new regularity condition for CR-mappings of nonminimal hypersurfaces, that we call \it Fuchsian type, \rm and prove its optimality for extension of CR-mappings to nonminimal points. \\ We also obtain an existence theorem for solutions of a class of singular complex ODEs (Theorem 3.5).

math.CV↗
arXiv2011

Holomorphic Extension of CR Functions from Quadratic Cones

It is proved that CR functions on a quadratic cone M in $\C^n$, n>1, admit one-sided holomorphic extension if and only if M does not have two-sided support, a geometric condition on M which generalizes minimality in the sense of Tumanov. A biholomorphic classification of quadratic cones in $\C^2$ is also given.

math.CV↗
arXiv2010

On the holomorphic closure dimension of real analytic sets

Given a real analytic (or, more generally, semianalytic) set R in the n-dimensional complex space, there is, for every point p in the closure of R, a unique smallest complex analytic germ X_p that contains the germ R_p. We call the complex dimension of X_p the holomorphic closure dimension of R at p. We show that the holomorphic closure dimension of an irreducible R is constant on the complement of a closed proper analytic subset of R, and discuss the relationship between this dimension and the CR dimension of R.

math.CV↗
arXiv2009

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.

math.CV↗
arXiv2008

Schlicht envelopes of holomorphy and foliations by lines

Given a domain Y in a complex manifold X, it is a difficult problem with no general solution to determine whether Y has a schlicht envelope of holomorphy in X, and if it does, to describe the envelope. The purpose of this paper is to tackle the problem with the help of a smooth 1-dimensional foliation F of X with no compact leaves. We call a domain Y in X an interval domain with respect to F if Y intersects every leaf of F in a nonempty connected set. We show that if X is Stein and if F satisfies a new property called quasiholomorphicity, then every interval domain in X has a schlicht envelope of holomorphy, which is also an interval domain. This result is a generalization and a global version of a well-known lemma from the mid-1980s. We illustrate the notion of quasiholomorphicity with sufficient conditions, examples, and counterexamples, and present some applications, in particular to a little-studied boundary regularity property of domains called local schlichtness.

math.CV↗
arXiv2007

Holomorphic correspondences between CR manifolds

It is proved that a germ of a real analytic CR map from a smooth real-analytic minimal CR manifold M to an essentially finite real-algebraic generic submanifold M' of P^N of the same CR-dimension extends as a holomorphic correspondence along M. Applications are given for pseudoconcave submanifolds of P^N.

math.CV↗