arXiv subjects

Vikramjeet Singh Chandel

Publications and source records attributed to Vikramjeet Singh Chandel.

15 recordsLinked to original sources

arXiv2026

On weak Wolff--Denjoy theorem for certain non-convex domains

In this paper, we provide a class of domains in $\mathbb{C}^3$, such that every holomorphic self-map of that domain either has a fixed point or the sequence of iterates is compactly divergent. In particular, it follows that the symmetrized bidisc, symmetrized tridisc, tetrablock, pentablock are in the aforementioned class of domains. We also prove that the fixed point set of a holomorphic self map of symmetrized bidisc and tetrablock is either empty set or a holomorphic retract. For the symmetrized bidisc, given a holomorphic self-map such that the sequence of iterates is compactly divergent, we also provide a description of its target set.

math.CV↗

arXiv2025

Geometry of horospheres in Kobayashi hyperbolic domains

For a Kobayashi hyperbolic domain, Abate introduced the notion of small and big horospheres of a given radius at a boundary point with a pole. In this article, we investigate which domains have the property that closed big horospheres and closed small horospheres centered at a given point and of a given radius intersect the boundary only at that point? We prove that any model-Gromov-hyperbolic domain have this property. To provide examples of non-Gromov-hyperbolic domains, we show that unbounded locally model-Gromov-hyperbolic domains and bounded, Dini-smooth, locally convex domains, that are locally visible, also have this property. Finally, using the geometry of the horospheres, we present a result about the homeomorphic extension of biholomorphisms and give an application of it.

math.CV↗

arXiv2025

Horofunction compactifications and local Gromov model domains

We explore the horofunction compactification of complete hyperbolic domains in complex Euclidean space equipped with the Kobayashi distance. We provide a sufficient condition under which, given a domain $Ω$ as above, the identity map from $Ω$ to itself extends to an embedding of $\overlineΩ$ into the horofunction compactification of $(Ω,k_Ω)$, with $k_Ω$ denoting the Kobayashi distance on $Ω$. Notably, this condition admits unbounded domains that are not Gromov hyperbolic relative to the Kobayashi distance. We also provide a large class of planar hyperbolic domains satisfying the above condition.

math.CV↗

arXiv2025

The metric compactification of a Kobayashi hyperbolic complex manifold and a Denjoy--Wolff Theorem

We study the metric compactification of a Kobayashi hyperbolic complex manifold $\mathcal{X} $ equipped with the Kobayashi distance $ \mathsf{k}_{\mathcal{X}} $. We show that this compactification is genuine -- i.e., $ \mathcal{X} $ embeds as a dense open subset -- even without completeness of $ \mathsf{k}_{\mathcal{X}} $, and that it becomes a \emph{good compactification} in the sense of Bharali--Zimmer when $(\mathcal{X}, \mathsf{k}_{\mathcal{X}}) $ is complete. As an application, we obtain a criterion for the continuous extension of quasi-isometric embeddings from $ (\mathcal{X}, \mathsf{k}_{\mathcal{X}}) $ into visibility domains of complex manifolds. For a Kobayashi hyperbolic domain $ Ω\subsetneq \mathcal{X} $, to each boundary point of $ Ω$ in the end compactification, we associate a fiber of metric boundary points. This allows the small and big horospheres of Abate to be expressed as the intersection and union of horoballs centered at metric boundary points. We use this to formulate a Wolff-type lemma in terms of horoballs and prove a Denjoy--Wolff theorem for complete hyperbolic domains satisfying a boundary divergence condition for $\mathsf{k}_{\mathcal{X}} $. Finally, we present necessary and sufficient conditions under which the identity map extends continuously between the metric and end compactifications.

math.CV↗

arXiv2024

Visibility property in one and several variables and its applications

In this paper we report our investigations on visibility with respect to the Kobayashi distance and its applications, with a special focus on planar domains. We prove that totally disconnected subsets of the boundary are removable in the context of visibility. We also show that a domain in $\mathbb{C}^n$ is a local weak visibility domain if and only if it is a weak visibility domain. The above holds also for visibility. Along the way, we prove an intrinsic localization result for the Kobayashi distance. Moreover, we observe some interesting consequences of weak visibility; for example, weak visibility implies compactness of the end topology of the closure of the domain. For planar domains: (i) We provide examples of visibility domains that are not locally Goldilocks at any boundary point. (ii) We provide certain general conditions on planar domains that yield the continuous extension of conformal maps, generalizing the Carathéodory extension theorem. Our conditions are quite general and assume very little regularity of the boundary. We demonstrate this through examples. (iii) We also provide conditions for the homeomorphic extension of biholomorphic maps up to the boundary. (iv) We prove that a hyperbolic, simply connected domain possesses the visibility property if and only if its boundary is locally connected. This leads us to reformulate the MLC conjecture in terms of visibility. (v) We provide a characterization of visibility for a large class of planar domains including certain uncountably connected domains.

math.CV↗

arXiv2023

Notions of Visibility with respect to the Kobayashi distance: Comparison and Applications

In this article, we study notions of visibility with respect to the Kobayashi distance for relatively compact complex submanifolds in Euclidean spaces. We present a sufficient condition for a domain to possess the visibility property relative to Kobayashi almost-geodesics introduced by Bharali--Zimmer (we call this simply the visibility property). As an application, we produce new classes of domains having this kind of visibility. Next, we introduce and study the notion of visibility subspaces of relatively compact complex submanifolds. Using this notion, we generalize to such submanifolds a recent result of Bracci--Nikolov--Thomas. The utility of this generalization is demonstrated by proving a theorem on the continuous extension of Kobayashi isometries. Finally, we prove a Wolff--Denjoy-type theorem that is a generalization of recent results of this kind by Bharali--Zimmer and Bharali--Maitra and that, owing to the new classes of domains mentioned, is a proper generalization. Along the way, we note that what is needed for the proof of this sort of theorem to work is a form of visibility that seems to be intermediate between what we are calling visibility and visibility with respect to ordinary Kobayashi geodesics.

math.CV↗

arXiv2021

On a spectral version of Cartan's theorem

For a domain $Ω$ in the complex plane, we consider the domain $S_n(Ω)$ consisting of those $n\times n$ complex matrices whose spectrum is contained in $Ω$. Given a holomorphic self-map $Ψ$ of $S_n(Ω)$ such that $Ψ(A)=A$ and the derivative of $Ψ$ at $A$ is identity for some $A\in S_n(Ω)$, we investigate when the map $Ψ$ would be spectrum-preserving. We prove that if the matrix $A$ is either diagonalizable or non-derogatory then for most domains $Ω$, $Ψ$ is spectrum-preserving on $S_n(Ω)$. Further, when $A$ is arbitrary, we prove that $Ψ$ is spectrum-preserving on a certain analytic subset of $S_n(Ω)$.

math.CV↗

arXiv2021

Irreducibility of a sum of polynomials depending on disjoint sets of variables

In this article, we give two different sufficient conditions for the irreducibility of a polynomial of more than one variable, over the field of complex numbers, that can be written as a sum of two polynomials which depend on mutually disjoint sets of variables. These conditions are derived from analyzing the Newton polytope of such a polynomial and then applying the `Irreducibility criterion' introduced by Gao.

math.AC↗

arXiv2021

Certain non-homogeneous matricial domains and Pick-Nevanlinna interpolation problem

In this article, we consider certain matricial domains that are naturally associated to a given domain of the complex plane. A particular example of such domains is the spectral unit ball. We present several results for these matricial domains. Our first result shows, generalizing a result of Ransford-White for the spectral unit ball, that the holomorphic automorphism group of these matricial domains does not act transitively. We also consider $2$-point and $3$-point Pick-Nevanlinna interpolation problem from the unit disc to these matricial domains. We present results providing necessary conditions for the existence of a holomorphic interpolant for these problems. In particular, we shall observe that these results are generalizations of the results provided by Bharali and Chandel related to these problems.

math.CV↗

arXiv2021

Interpolating sequences for the Banach algebras generated by a class of test functions

Given a domain $Ω$ in $\mathbb{C}^n$ and a collection of test functions $Ψ$ on $Ω$, we consider the complex-valued $Ψ$-Schur-Agler class associated to the pair $(Ω,\,Ψ)$. In this article, we characterize interpolating sequences for the associated Banach algebra of which the $Ψ$-Schur-Agler class is the closed unit ball. When $Ω$ is the unit disc $\mathbb{D}$ in the complex plane $\mathbb{C}$ and the class of test function includes only the identity function on $\mathbb{D}$, the aforementioned algebra is the algebra of bounded holomorphic functions on $\mathbb{D}$ and in this case, our characterization reduces to the well known result by Carleson. Furthermore, we present several other cases of the pair $(Ω,\,Ψ)$, where our main result could be applied to characterize interpolating sequences which also show the efficacy of our main result.

math.FA↗

arXiv2021

On the Nevanlinna problem - Characterization of all Schur-Agler class solutions

Given a domain $Ω$ in $\mathbb{C}^m$, and a finite set of points $z_1,\ldots, z_n\in Ω$ and $w_1,\ldots, w_n\in \mathbb{D}$ (the open unit disc in the complex plane), the $Pick\, interpolation\, problem$ asks when there is a holomorphic function $f:Ω\rightarrow \overline{\mathbb{D}}$ such that $f(z_i)=w_i,1\leq i\leq n$. Pick gave a condition on the data $\{z_i, w_i:1\leq i\leq n\}$ for such an $interpolant$ to exist if $Ω=\mathbb{D}$. Nevanlinna characterized all possible functions $f$ that $interpolate$ the data. We generalize Nevanlinna's result to an arbitrary set $Ω$. In this case, the function $f$ comes from the Schur-Agler class. The abstract result is then applied to three examples - the bidisc, the symmetrized bidisc and the annulus. In these examples, the Schur-Agler class is the same as the Schur class.

math.FA↗

arXiv2019

On the Nevanlinna problem -- Characterization of all Schur-Agler class solutions affiliated with a given kernel

Given a domain $Ω$ in $\mathbb{C}^m$, and a finite set of points $z_1,z_2,\ldots, z_n\in Ω$ and $w_1,w_2,\ldots, w_n\in \mathbb{D}$ (the open unit disc in the complex plane), the \textit{Pick interpolation problem} asks when there is a holomorphic function $f:Ω\rightarrow \overline{\mathbb{D}}$ such that $f(z_i)=w_i,1\leq i\leq n$. Pick gave a condition on the data $\{z_i, w_i:1\leq i\leq n\}$ for such an $interpolant$ to exist if $Ω=\mathbb{D}$. Nevanlinna characterized all possible functions $f$ that \textit{interpolate} the data. We generalize Nevanlinna's result to a domain $Ω$ in $\mathbb{C}^m$ admitting holomorphic test functions when the function $f$ comes from the Schur-Agler class and is affiliated with a certain completely positive kernel. The Schur class is a naturally associated Banach algebra of functions with a domain. The success of the theory lies in characterizing the Schur class interpolating functions for three domains - the bidisc, the symmetrized bidisc and the annulus - which are affiliated to given kernels.

math.FA↗

arXiv2017

The 3-point spectral Pick interpolation problem and an application to holomorphic correspondences

We provide a necessary condition for the existence of a 3-point holomorphic interpolant $F:\mathbb{D}\longrightarrowΩ_n$, $n\geq 2$. Our condition is inequivalent to the necessary conditions hitherto known for this problem. The condition generically involves a single inequality and is reminiscent of the Schwarz lemma. We combine some of the ideas and techniques used in our result on the $\mathcal{O}(\mathbb{D},Ω_n)$-interpolation problem to establish a Schwarz lemma -- which may be of independent interest -- for holomorphic correspondences from $\mathbb{D}$ to a general planar domain $Ω\Subset \mathbb{C}$.

math.CV↗

arXiv2016

Pick interpolation on the polydisc: small families of sufficient kernels

We give a solution to Pick's interpolation problem on the unit polydisc in $\mathbb{C}^n$, $n\geq 2$, by characterizing all interpolation data that admit a $\mathbb{D}$-valued interpolant, in terms of a family of positive-definite kernels parametrized by a class of polynomials. This uses a duality approach that has been associated with Pick interpolation, together with some approximation theory. Furthermore, we use duality methods to understand the set of points on the $n$-torus at which the boundary values of a given solution to an extremal interpolation problem are not unimodular.

math.CV↗

arXiv2016

The three-point Pick-Nevanlinna interpolation problem on the polydisc

We give a characterization for the existence of a holomorphic interpolant on the unit polydisc $\mathbb{D}^n,$ $n\geq 2,$ for prescribed three-point Pick--Nevanlinna data. One of the key steps is a characterization for the existence of an interpolant that is a rational inner function on $\mathbb{D}^n.$ The latter reduces the search for a three-point interpolant to finding a single rational inner function that satisfies a type of positivity condition and arises from a polynomial of a very special form. This in turn relies on a pair of results, which are of independent interest, on the factorization of rational inner functions.

math.CV↗