On dicritical singularities of Levi-flat set
S. Pinchuk, R. Shafikov, and A. Sukhov • Ark. Mat., 56 (2018), no. 2, 395–408.
Abstract
It is proved that dicritical singularities of real analytic Levi-flat sets coincide
with the set of Segre degenerate points.
Uniformization and Steinness
S. Nemirovski and R. Shafikov • Canad. Math. Bull. 61 (2018), no. 3, 637–639.
Abstract
It is shown that the unit ball in $\mathbb C^n$ is the only complex manifold that can
universally cover both Stein and non-Stein strictly pseudoconvex domain.
Some aspects of holomorphic mappings: a survey
S. Pinchuk, R. Shafikov, A. Sukhov • Proc. Steklov Inst. Math. 298 (2017), no. 1, 212–247.
Abstract
This expository paper is concerned with the properties of proper holomorphic mappings between domains in complex affine spaces. We discuss some of the main geometric methods of this theory, such as the reflection principle, the scaling method, and the Kobayashi–Royden metric. We sketch the proofs of certain principal results and discuss some recent
achievements. Several open problems are also stated
Dicritical singularities and laminar currents on Levi-flat hypersurfaces
S. Pinchuk, R. Shafikov, and A. Sukhov • Izv. Math. 81 (2017), no. 5, 1030–1043.
Abstract
We establish an effective criterion for a dicritical singularity of a real analytic Levi-flat hypersurface. The criterion is stated in terms of Segre varieties. As an application, we obtain a structure theorem for a certain class of currents in the non-dicritical case.
Rational and Polynomial Density on Compact Real Manifolds
P. Gupta and R. Shafikov • Internat. J. Math. Vol. 28, No. 5 (2017), 1750040.
Abstract
We establish a characterization for an $m$-manifold $M$ to admit $n$ functions $f_1,\dots,f_n$ and
$n′$ functions $g_1,\dots, g_{n′}$ in $C^\infty(M)$ so that every element of $C^k(M)$ can be approximated
by rational combinations of $f_1,\dots,f_n$ and polynomial combinations of $g_1,\dots,{g_{n′}}$ . As
an application, we show that the optimal value of $n$ and $n′$ for all manifolds of dimension
$m$ is $[3m/2]$, when $k \ge 1$ and $m \ge 2$.
Discs in hulls of real immersions to Stein manifolds
R. Shafikov, A. Sukhov • Proc. Steklov Inst. Math. Vol 298, pp. 334 - 344, 2017.
Abstract
We obtain results on the existence of complex discs in plurisubharmonically convex
hulls of Lagrangian and totally real immersions into Stein manifolds
Distributional boundary values of holomorphic functions on product domains
D. Chakrabarti and R. Shafikov • Math. Z. 286 (2017), no. 3-4, 1145–1171.
Abstract
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 also prove an analog of the Bochner–Hartogs theorem for these
boundary values for the simplest such domains, the product domains
Distributional boundary values: some new perspectives
D. Chakrabarti and R. Shafikov • Contemp. Math. 681 (2017), AMS, 65-70.
Abstract
It is shown that there is a complex manifold $M$, a piecewise smooth domain $\Omega \subset M$ and a holomorphic function $f$ of polynomial growth on $\Omega$ such that the boundary current $\text{bc}f$ induced by $f$ does not exist.
Rational approximation and Lagrangian inclusions
R. Shafikov, A. Sukhov • Enseign. Math. 62 (2016), no. 3-4, 487–499.
Abstract
We show that any real compact surface $S$ , except the sphere $S^2$ and the projective
plane $\mathbb{RP}^2$ , admits a pair of smooth complex-valued functions $f_1$ , $f_2$ with the property
that any continuous complex-valued function on $S$ is a uniform limit of a sequence
$\{R_j(f_1,f_2)\}$ , where $R_j(z_1,z_2)$ are rational functions on $\mathbb C_2$.
Open Whitney umbrellas are locally polynomially convex
O. Mitrea and R. Shafikov • Proc. Amer. Math. Soc. 144 (2016), no. 12, 5319–5332.
Abstract
It is proved that any smooth open Whitney umbrella in $\mathbb C^2$ is locally polynomially convex near the singular point.
Lagrangian inclusion with an open Whitney umbrella is rationally convex
R. Shafikov, A. Sukhov • Contemp. Math., 662, Amer. Math. Soc., 2016.
Abstract
It is shown that a Lagrangian inclusion of a real surface in $\mathbb C^2$ with a standard open
Whitney umbrella and double transverse self-intersections is rationally convex.
Polynomially convex hulls of singular real manifolds.
R. Shafikov, A. Sukhov • Trans. Amer. Math. Soc. 368 (2016), no. 4, 2469–2496.
Abstract
We obtain local and global results on polynomially convex hulls of Lagrangian and totally real submanifolds
of $\mathbb{C}^n$ with self-intersections and open Whitney umbrella points.
Divergent CR-Equivalences and Meromorphic Differential Equations
I. Kossovskiy and R. Shafikov • J. Eur. Math. Soc. (JEMS) 18 (2016), no. 12, 2785–2819.
Abstract
Using the analytic theory of differential equations, we construct, in any positive CR-dimension and CR-codimension, examples of formally but not holomorphically equivalent real-analytic CR-submanifolds in complex spaces.
Analytic Differential Equations and Spherical Real Hypersurfaces
I. Kossovskiy and R. Shafikov • J. Differential Geom. 102 (2016), no. 1, 67–126.
Abstract
We establish an injective correspondence $M \to E(M)$ 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 $\dim \mathfrak{hol}(M, p) \le 5$ for the infinitesimal automorphism algebra of an arbitrary 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 proof of the dimension gap $\dim \mathfrak{hol}(M, p) = \{8, 5, 4, 3, 2, 1, 0\}$ 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 Fuchsian type, and prove its optimality for the extension of CR-mappings to nonminimal points. We also obtain an existence theorem for solutions of a class of singular complex ODEs.
Critical sets of proper holomorphic mappings
S. Pinchuk and R. Shafikov • Proc. Amer. Math. Soc. 143 (2015), no. 10, 4335–4345.
Abstract
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 $M$ 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 $\mathbb C^n$.
Germs of singular Levi-flat hypersurfaces and holomorphic foliations
R. Shafikov and A. Sukhov • Comment. Math. Helv. 90 (2015), no. 2, 479 – 502.
Abstract
It is shown that the Levi foliation of a real analytic Levi-flat hypersurface extends
to a $d$-web near a nondicritical singular point and admits a multiple-valued meromorphic first
integral.
Tameness of complex dimension in real analytic sets
Adamus, J., Randriambololona, S., Shafikov, R. • Canadian J. Math., 65 (2013), no. 4, 721–739.
Abstract
Given a real analytic set $X$ in a complex manifold and a positive integer $d$, denote by $A^d$ the
set of points $p$ in $X$ at which there exists a germ of a complex analytic set of dimension $d$ contained
in $X$. It is proved that $A^d$ is a closed semianalytic subset of $X$.
Holomorphic mappings between domains in \(\mathbb{C}^2\)
Shafikov, R., Verma, K. • Canad. J. Math. 64(2), 2012, pp. 429–454.
Abstract
An extension theorem for holomorphic mappings between two domains in $\mathbb C^2$ is proved
under purely local hypotheses
On the holomorphic closure dimension of real analytic sets
Adamus, J., Shafikov, R. • Trans. Amer. Math. Soc. 363 (2011), no 11, 5761-5772.
Abstract
Given a real analytic (or, more generally, semianalytic) set $R$ in $\mathbb{C}^n$ (viewed as $\mathbb R^{2n}$), there is, for every $p \in \overline R$, a unique smallest complex analytic germ $X_p$ that contains the germ $R_p$. We call $\dim_{\mathbb C} X_p$ 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 we discuss the relationship between this dimension and the CR-dimension of $R$.
CR functions on Subanalytic Hypersurfaces
Chakrabarti, D., Shafikov, R. • Indiana Univ. Math. J. 59 No. 2 (2010), 459–494.
Abstract
A general class of singular real hypersurfaces, called subanalytic, is defined. For a subanalytic hypersurface
$M$ in $\mathbb C^n$, 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.
