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Golam Mostafa Mondal

Publications and source records attributed to Golam Mostafa Mondal.

7 recordsLinked to original sources
arXiv2026

Certain real surfaces in $\mathbb{C}^2$ with degenerated CR singularities

In this paper, we study the local polynomial convexity of certain smooth real surfaces in \(\mathbb{C}^2\) with isolated CR singularity at the origin with higher-order of degeneracy. Under the assumption that the surface can be pulled back to a union of finitely many pairwise transverse totally real surfaces by a proper holomorphic map from $\mathbb{C}^2$ to $\mathbb{C}^2$, we obtain a normal form for such surfaces near the origin as $\{(z,w)\in\mathbb{C}^2: w= \overline{z}^k+o(|z|^{k})\}$ or $M_t := \left\{ (z,w)\in\mathbb{C}^2 : w=(z+t\overline{z})^k+o(|z|^k) \right\}$, for some \(t>0\), where the parameter $t$ is a local biholomorphic invariant. We focus on the surfaces with order of degeneracy $k\geq 3$. We prove that $M_t$ is locally polynomially convex at the origin if $t>cosec\left(\fracπ{k}\right)$. On the other hand, for $0<t<\frac{1}{k-2}$, we will also show that $M_t$ fails to be locally polynomially convex at the origin; and furthermore, a $(2k-3)$-parameter family of analytic discs attached to $M_t$ for $0<t<\min\left\{\sin\left(\fracπ{k}\right),\frac{1}{k-2}\right\}$.

math.CV↗
arXiv2025

Uniform approximation on certain polynomial polyhedra in $\mathbb{C}^2$

In this paper we extend the dichotomy given by Samuelsson and Wold that can be thought of as an analogue of the Wermer maximality theorem in $\mathbb{C}^2$ for certain polynomial polyhedra. We consider complex non-degenerate simply connected polynomial polyhedra of the form $Ω:=\{z\in\mathbb{C}^2: |p_1(z)|<1, |p_2(z)|<1\}$ such that $\overlineΩ$ is compact. Under a mild condition of the polynomials $p_1$ and $p_2$, we prove that either the uniform algebra, generated by polynomials and some continuous functions $f_1,\dots, f_N$ on the distinguished boundary that extends as pluriharmonic functions on $Ω$, is all continuous functions on the distinguished boundary or there exists an algebraic variety in $\overlineΩ$ on which each $f_j$ is holomorphic. We also compute the polynomial hull of the graph of pluriharmonic functions in some cases where the pluriharmonic functions are conjugates of holomorphic polynomials. We also give a couple of general theorem about uniform approximation on the domains with low boundary regularity.

math.CV↗
arXiv2024

On Hyperbolicity of Spirallike Circularlike domain

In this paper, we prove that a spirallike circularlike domain is Kobayashi hyperbolic if and only if its core is empty. In particular, we show that such a domain is Kobayashi hyperbolic if and only if it is (biholomorphic to) a bounded domain. We also propose a problem in this area.

math.CV↗
arXiv2023

Uniform algebras and distinguished varieties

In this article, we point out the connections between the distinguished varieties introduced by Agler and McCarthy with certain uniform algebras on bidisc studied by Samuelsson and Wold. We also prove analogues of Samuelsson-Wold result for the domains in $\mathbb{C}^2$ that are the images of the bidisc under certain proper polynomial map on $\mathbb{C}^2$. We also give a description of polynomial convex hull of graph of anti-holomorphic polynomial over the distinguished boundary of such domains. We mention the case for the symmetrized bidisc as an example.

math.CV↗
arXiv2023

Polynomial convexity of compacts that lies in certain Levi-flat hypersurfaces in $\mathbb{C}^2$

In this paper, we first prove that the totally real discs lying in certain Levi flat hypersurfaces are polynomially convex. As applications we prove that the totally real discs lying in the boundary of certain polynomial polyhedra are polynomially convex. We also provide an if and only if condition for polynomial convexity of totally real discs lying in the boundary of Hartog's triangle. We also provide sufficient conditions on general compact subsets lying on those hypersurfaces for polynomial convexity.

math.CV↗
arXiv2022

Polynomial Convexity and Polynomial approximations of certain sets in $\mathbb{C}^{2n}$ with non-isolated CR-singularities

In this paper, we first consider the graph of $(F_1,F_{2},\cdots,F_{n})$ on $\overline{\mathbb{D}}^{n},$ where $F_{j}(z)=\bar{z}^{m_{j}}_{j}+R_{j}(z),j=1,2,\cdots,n,$ which has non-isolated CR-singularities if $m_{j}>1$ for some $j\in\{1,2,\cdots,n\}.$ We show that under certain condition on $R_{j},$ the graph is polynomially convex and holomorphic polynomials on the graph approximates all continuous functions. We also show that there exists an open polydisc $D$ centred at the origin such that the set $\{(z^{m_{1}}_{1},\cdots, z^{m_{n}}_{n}, \bar{z_1}^{m_{n+1}} + R_{1}(z),\cdots, \bar{z_{n}}^{m_{2n}} + R_{n}(z)):z\in \overline{D},m_{j}\in \mathbb{N}, j=1,\cdots,2n\}$ is polynomially convex; and if $\gcd(m_{j},m_{k})=1~~\forall j\not=k,$ the algebra generated by the functions $z^{m_{1}}_{1},\cdots, z^{m_{n}}_{n}, \bar{z_1}^{m_{n+1}} + R_{1},\cdots, \bar{z_{n}}^{m_{2n}} + R_{n}$ is dense in $\mathcal{C}(\overline{D}).$ We prove an analogue of Minsker's theorem over the closed unit polydisc, i.e, if $\gcd(m_{j},m_{k})=1~~\forall j\not=k,$ the algebra $[z^{m_{1}}_{1},\cdots, z^{m_{n}}_{n}, \bar{z_1}^{m_{n+1}},\cdots , \bar{z_{n}}^{m_{2n}};{\overline{\mathbb{D}}^{n}} ]=\mathcal{C}(\overline{\mathbb{D}}^{n}).$ In the process of proving the above results, we also studied the polynomial convexity and approximation of certain graphs.

math.CV↗