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Amar Deep Sarkar

Publications and source records attributed to Amar Deep Sarkar.

12 recordsLinked to original sources
arXiv2026

A Canonical Positive Definite Kernel Associated with the $\xi$-Bergman Kernel

Let $\Omega \subset \mathbb{C}^{n}$ and $\xi \in \ell^{1}$. The $\xi$-Bergman kernel $K_{\xi, \Omega}$, introduced by Bao and Guan, generalizes the classical Bergman kernel by replacing the point evaluation functional with a functional determined by sequence $\xi$. While this kernel inherits several important extremal and plurisubharmonic properties, it is intrinsically an on-diagonal object and therefore lacks the two-variable reproducing kernel structure that lies at the heart of the classical Bergman theory. The purpose of this paper is to associate a canonical Hermitian positive-definite kernel with the $\xi$-Bergman kernel and to investigate its analytic and geometric properties. Our construction is based on the family of Riesz representatives corresponding to the $\xi$-evaluation functionals. More precisely, we introduce a Hermitian kernel obtained as the Gram kernel of these representatives and show that it is positive definite and for $z \in \Omega$ satisfies \[ B_{\xi,\Omega}(z,z)=K_{\xi,\Omega}(z), \] thereby recovering the $\xi$-Bergman kernel as its diagonal restriction. As a consequence, we prove that the $\xi$-Bergman kernel is real analytic on $\Omega$. We also establish biholomorphic transformation laws, and obtain a representation of the $\xi$-Bergman kernel in terms of derivatives of the classical Bergman kernel. Furthermore, we obtain explicit formulas for the $\xi$-Bergman kernel on the upper half-plane $\mathbb{H}$ corresponding to several classes of sequences $\xi$, establish corresponding $\xi$-Lu Qi-Keng results, and derive precise boundary asymptotics. These examples illustrate how the choice of the differential functional influences both the zero set and the boundary growth of the associated kernel.

math.CV↗
arXiv2026

On the exponential convergence of Kobayashi geodesics in strongly convex domains

In this paper, we have proved a quantitative version of the approaching geodesic property for certain convex domains. We have proved that that if $\Omega \subset \mathbb{C}^{d}$ is a bounded strongly convex domain with $\mathcal{C}^3$ boundary and $\gamma_{1}, \gamma_{2}:[0, \infty) \to \Omega$ are two geodesic rays such that $\gamma_{1}(\infty)=\gamma_{2}(\infty)=\xi \in \partial \Omega$. Then if the images of $\gamma_{1}$ and $\gamma_{2}$ are contained in the same complex geodesic, then there exists $T\in \mathbb{R}$ \[ \lim_{t \to \infty} \frac{1}{t} \log K_{\Omega}\big(\gamma_{1}(t), \gamma_{2}(t+T)\big) = -2, \] otherwise \[ \lim_{t \to \infty} \frac{1}{t} \log K_{\Omega}\big(\gamma_{1}(t), \gamma_{2}(t+T)\big) = -1. \] Furthermore, using this property we provided a characterization of strongly pseudoconvex domain via a biholomorphic invariant function namely generalized squeezing function. We have proved that: For every $\alpha>0$ there exists $\epsilon(d,\alpha)>0$ such that the following holds: if $\Omega \subset \mathbb{C}^d$ is a bounded convex domain with $\mathcal{C}^{2,\alpha}$-boundary and \[ T_{\Omega}^{D}(z)\geq 1-\epsilon \] outside a compact subset of $\Omega$, where $D \Subset \mathbb{C}^{d}$ is a balanced strongly convex domain with $\mathcal{C}^{3}$ boundary and $T_{\Omega}^{D}$ is the squeezing function of $\Omega$ with respect to the domain $D$ then $\Omega$ is strongly pseudoconvex. We also establish exponential convergence of a certain family of quasi-geodesics in the unit ball of $\mathbb{C}^{d}$. We further show that the study of this family of quasi-geodesics provides a useful tool that allows the exponential convergence property of geodesics to be transferred from local subdomains to the ambient domain, as well as in the reverse direction.

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↗
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↗
arXiv2024

A Note on Kernel Functions of Dirichlet Spaces

For a planar domain $Ω$, we consider the Dirichlet spaces with respect to a base point $ζ\inΩ$ and the corresponding kernel functions. It is not known how these kernel functions behave as we vary the base point. In this note, we prove that these kernel functions vary smoothly. As an application of the smoothness result, we prove a Ramadanov-type theorem for these kernel functions on $Ω\timesΩ$. This extends the previously known convergence results of these kernel functions. In fact, we have made these observations in a more general setting, that is, for weighted kernel functions and their higher-order counterparts.

math.CV↗
arXiv2023

The Reduced Bergman Kernel and its Properties

In this article, we study some properties of the $n$-th order weighted reduced Bergman kernels for planar domains, $n\geq 1$. Specifically, we look at Ramadanov type theorems, localization, and boundary behaviour of the weighted reduced Bergman kernel and its higher-order counterparts. We also give a transformation formula for these kernels under biholomorphisms.

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↗
arXiv2023

Localization of the Kobayashi distance for any visibility domain

In this article, we prove localization results for the Kobayashi distance of Kobayashi hyperbolic domains with local visibility property in $\mathbb{C}^d$, $d \geq 1$. This is done by proving a localization result for the Kobayashi-Royden pseudometric, along with some other results for domains satisfying local weak visibility.

math.CV↗
arXiv2022

Transformation formula for the Reduced Bergman kernel and its Application

In this article, we prove the transformation formula for the reduced Bergman kernels under proper holomorphic correspondences between bounded domains in the complex plane. As a corollary, we obtain the transformation formula for the reduced Bergman kernels under proper holomorphic maps. We also establish the transformation formula for the weighted reduced Bergman kernels under proper holomorphic maps. Finally, we provide an application of this transformation formula.

math.CV↗
arXiv2020

Boundary behaviour of the Span metric and its higher-order curvatures

In this note, we use scaling principle to study the boundary behaviour of the span metric and its higher-order curvatures on finitely connected Jordan planar domains. A localization of this metric near boundary points of finitely connected Jordan domains is also obtained. Further, we obtain boundary sharp estimates for this metric on $ C^2 $-smooth bounded domains and consequently, this metric is comparable to the Carathéodory and the Kobayashi metrics on these domains.

math.CV↗
arXiv2019

Boundary behaviour of some conformal invariants on planar domains

The purpose of this note is to use the scaling principle to study the boundary behaviour of some conformal invariants on planar domains. The focus is on the Aumann--Carathéodory rigidity constant, the higher order curvatures of the Carathéodory metric and two conformal metrics that have been recently defined.

math.CV↗