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Kaushal Verma

Publications and source records attributed to Kaushal Verma.

At least 19 recordsLinked to original sources
arXiv2026

A quasiconformal variant of the union problem

The Union Problem, which has its genesis in the classical Levi problem, asks for a classification of complex manifolds $M$ that can be exhausted by an increasing union of submanifolds $M_j \subset M$ which are all biholomorphic to a fixed domain in $\mathbb C^n$. We explore a quasiconformal variant of this question and seek to classify $n$-Riemannian manifolds $M$ such that each $M_j$ is quasiconformally equivalent to a bounded domain in $\mathbb R^n$. It turns out that this is possible when these quasiconformal equivalences have uniformly bounded dilatations. Using Kiernan's quasiconformal Schwarz lemma when $n=2$ and Ferrand's conformal capacity when $n \geq 3$, we classify a class of $n$-Riemannian manifolds $M$ such that each $M_j$ is $K_j$-quasiconformally equivalent to $\Omega \setminus A$, where $\sup K_j < \infty$ and $\Omega \subset \mathbb R^n$ is a $C^2$-smoothly bounded domain and $A \subset \Omega$ is at most finite. As a consequence, we obtain that Gehring's example of a bounded domain in $\mathbb R^n$ which has $C^1$-smooth boundary everywhere except at a point and is known to be quasiconformally inequivalent to the unit ball in $\mathbb R^n$, possesses the additional property that it cannot even be exhausted by quasiconformal images of the unit ball with uniformly bounded dilatations.

math.CV↗
arXiv2026

Rigidity of the escaping set of polynomial automorphisms of $\mathbb{C}^2$

Let $H$ be a polynomial automorphism of $\mathbb{C}^2$ of positive entropy and degree $d \ge 2$. We prove that the escaping set $U^+$ (or equivalently, the non-escaping set $K^+$), of $H$ is rigid under the action of holomorphic automorphisms of $\mathbb{C}^2$. Specifically, every holomorphic automorphism of $\mathbb{C}^2$ that preserves $U^+$ essentially takes the form $L \circ H^s$ where $s \in \mathbb{Z}$ and $L$ belongs to a finite cyclic group of affine maps that preserve the escaping set. Second, note that the sub-level sets $\{G^+ < c\}$, $c > 0$, of the Greens function $G^+$ associated with the map $H$ are canonical examples of Short $\mathbb{C}^2$s. As a consequence of the above theorem, we show that the holomorphic automorphisms of these Short $\mathbb{C}^2$s are affine automorphisms of $\mathbb{C}^2$ preserving the escaping set $U^+$. Hence, the automorphism group of these Short $\mathbb{C}^2$s are the same for every $c>0$ and is a finite cyclic group.

math.CV↗
arXiv2025

Weighted Kernel Functions on Planar Domains

We study the variation of weighted Szegő and Garabedian kernels on planar domains as a function of the weight. A Ramadanov type theorem is shown to hold as the weights vary. As a consequence, we derive properties of the zeros of the weighted Szegő and Garabedian kernel for weights close to the constant function $1$ on the boundary. We further study the weighted Ahlfors map and strengthen results concerning its boundary behaviour. Explicit examples of the weighted kernels are presented for certain classes of weights. We highlight an interesting property of the weighted Szegő and Garabedian kernels, implicit in Nehari's work, and explore several of its consequences. Finally, we discuss the weighted Carathéodory metric, and describe relations of the weighted Szegő and Garabedian kernel with certain classical kernel functions.

math.CV↗
arXiv2025

Weighted Szegő Kernels on Planar Domains

We study properties of weighted Szegő and Garabedian kernels on planar domains. Motivated by the unweighted case as explained in Bell's work, the starting point is a weighted Kerzman-Stein formula that yields boundary smoothness of the weighted Szegő kernel. This provides information on the dependence of the weighted Szegő kernel as a function of the weight. When the weights are close to the constant function $1$ (which corresponds to the unweighted case), it is shown that some properties of the unweighted Szegő kernel propagate to the weighted Szegő kernel as well. Finally, it is shown that the reduced Bergman kernel and higher order reduced Bergman kernels can be written as a rational combination of three unweighted Szegő kernels and their conjugates, thereby extending Bell's list of kernel functions that are made up of simpler building blocks that involve the Szegő kernel.

math.CV↗
arXiv2024

Quasiconformal variants of the Wong--Rosay theorem

The Wong--Rosay theorem provides a characterization of the unit ball among all strongly pseudoconvex domains in terms of holomorphic automorphism group actions. We explore variants of this theorem in the quasiconformal setting.

math.CV↗
arXiv2024

Uniform non-autonomous basins of attraction

It has been conjectured that every stable manifold arising from a holomorphic automorphism, that acts hyperbolically on a compact invariant set, is biholomorphic to complex Euclidean space. Such stable manifolds are known to be biholomorphic to the basin of a uniformly attracting family of holomorphic maps. It is shown that the basin of a uniformly attracting family of holomorphic maps is biholomorphic to complex Euclidean space and this resolves the conjecture on the biholomorphism type of such stable manifolds affirmatively.

math.CV↗
arXiv2024

Weighted Bergman Kernels on Planar Domains

Boundary Behaviour of Weighted Bergman Kernels: For a planar domain $D \subset \mathbb{C}$ and an admissible weight function $μ$ on it, some aspects of the boundary behaviour of the corresponding weighted Bergman kernel $K_{D, μ}$ are studied. First, under the assumption that $μ$ extends continuously to a smooth boundary point $p$ of $D$ and is non-vanishing there, we obtain a precise relation between $K_{D, μ}$ and the classical Bergman kernel $K_D$ near $p$. Second, when viewed as functions of such weights, the weighted Bergman kernel is shown to have a suitable additive and multiplicative property near such boundary points. A Study on Holomorphic Isometries of Weighted Bergman Metrics: For a domain $D \subset \mathbb{C}^n$ and an admissible weight $μ$ on it, we consider the weighted Bergman kernel $K_{D, μ}$ and the corresponding weighted Bergman metric on $D$. In particular, motivated by work of Mok, Ng, Chan--Yuan and Chan--Xiao--Yuan among others, we study the nature of holomorphic isometries from the disc $\mathbb{D} \subset \mathbb{C}$ with respect to the weighted Bergman metrics arising from weights of the form $μ= K_{\mathbb{D}}^{-d}$ for some integer $d \geq 0$. These metrics provide a natural class of examples that give rise to positive conformal constants that have been considered in various recent works on isometries. Specific examples of isometries that are studied in detail include those in which the isometry takes values in $\mathbb{D}^n$ and $\mathbb{D} \times \mathbb{B}^n$ where each factor admits a weighted Bergman metric as above for possibly different non-negative integers $d$. Finally, the case of isometries between polydisks in possibly different dimensions, in which each factor has a different weighted Bergman metric as above, is also presented.

math.CV↗
arXiv2023

Regularity of the leafwise Poincare metric on singular holomorphic foliations

Let $\mathcal F$ be a smooth Riemann surface foliation on $M \setminus E$, where $M$ is a complex manifold and the singular set $E \subset M$ is an analytic set of codimension at least two. Fix a hermitian metric on $M$ and assume that all leaves of $\mathcal F$ are hyperbolic. Verjovsky's modulus of uniformization $η$ is a positive real function defined on $M \setminus E$ defined in terms of the family of holomorphic maps from the unit disc $\mathbb D$ into the leaves of $\mathcal F$ and is a measure of the largest possible derivative in the class of such maps. Various conditions are known that guarantee the continuity of $η$ on $M \setminus E$. The main question that is addressed here is its continuity at points of $E$. To do this, we adapt Whitney's $C_4$-tangent cone construction for analytic sets to the setting of foliations and use it to define the tangent cone of $\mathcal F$ at points of $E$. This leads to the definition of a foliation that is of {\it transversal type} at points of $E$. It is shown that the map $η$ associated to such foliations is continuous at $E$ provided that it is continuous on $M \setminus E$ and $\mathcal F$ is of transversal type. We also present observations on the locus of discontinuity of $η$. Finally, for a domain $U \subset M$, we consider $\mathcal F_U$, the restriction of $\mathcal F$ to $U$ and the corresponding positive function $η_U$. Using the transversality hypothesis leads to strengthened versions of the results of Lins Neto--Martins on the variation $U \mapsto η_U$.

math.CV↗
arXiv2022

On the automorphism group of certain Short $\mathbb C^2$'s

For a Hénon map of the form $H(x, y) = (y, p(y) - ax)$, where $p$ is a polynomial of degree at least two and $a \not= 0$, it is known that the sub-level sets of the Green's function $G^+_H$ associated with $H$ are Short $\mathbb C^2$'s. For a given $c > 0$, we study the holomorphic automorphism group of such a Short $\mathbb C^2$, namely $Ω_c = \{ G^+_H < c \}$. The unbounded domain $Ω_c \subset \mathbb C^2$ is known to have smooth real analytic Levi-flat boundary. Despite the fact that $Ω_c$ admits an exhaustion by biholomorphic images of the unit ball, it turns out that its automorphism group, Aut$(Ω_c)$ cannot be too large. On the other hand, examples are provided to show that these automorphism groups are non-trivial in general. We also obtain necessary and sufficient conditions for such a pair of Short $\mathbb C^2$'s to be biholomorphic.

math.CV↗
arXiv2021

Narasimhan--Simha type metrics on strongly pseudoconvex domains in $\mathbb{C}^n$

For a bounded domain $D \subset \mathbb{C}^n$, let $K_D = K_D(z) > 0$ denote the Bergman kernel on the diagonal and consider the reproducing kernel Hilbert space of holomorphic functions on $D$ that are square integrable with respect to the weight $K_D^{-d}$, where $d \geq 0$ is an integer. The corresponding weighted kernel $K_{D, d}$ transforms appropriately under biholomorphisms and hence produces an invariant Kähler metric on $D$. Thus, there is a hierarchy of such metrics starting with the classical Bergman metric that corresponds to the case $d=0$. This note is an attempt to study this class of metrics in much the same way as the Bergman metric has been with a view towards identifying properties that are common to this family. When $D$ is strongly pseudoconvex, the scaling principle is used to obtain the boundary asymptotics of these metrics and several invariants associated to them. It turns out that all these metrics are complete on strongly pseudoconvex domains.

math.CV↗
arXiv2021

Limits of an increasing sequence of complex manifolds

Let $M$ be a complex manifold which admits an exhaustion by open subsets $M_j$ each of which is biholomorphic to a fixed domain $Ω\subset \mathbb C^n$. The main question addressed here is to describe $M$ in terms of $Ω$. Building on work of Fornaess--Sibony, we study two cases namely, $M$ is Kobayashi hyperbolic and the other being the corank one case in which the Kobayashi metric degenerates along one direction. When $M$ is Kobayashi hyperbolic, its complete description is obtained when $Ω$ is one of the following domains -- (i) a smoothly bounded Levi corank one domain, (ii) a smoothly bounded convex domain, (iii) a strongly pseudoconvex polyhedral domain in $\mathbb C^2$, or (iv) a simply connected domain in $\mathbb C^2$ with generic piecewise smooth Levi-flat boundary. With additional hypotheses, the case when $Ω$ is the minimal ball or the symmetrized polydisc in $\mathbb C^n$ can also be handled. When the Kobayashi metric on $M$ has corank one and $Ω$ is either of (i), (ii) or (iii) listed above, it is shown that $M$ is biholomorphic to a locally trivial fibre bundle with fibre $\mathbb C$ over a holomorphic retract of $Ω$ or that of a limiting domain associated with it. Finally, when $Ω= Δ\times \mathbb B^{n-1}$, the product of the unit disc $Δ\subset \mathbb C$ and the unit ball $\mathbb B^{n-1} \subset \mathbb C^{n-1}$, a complete description of holomorphic retracts is obtained. As a consequence, if $M$ is Kobayashi hyperbolic and $Ω= Δ\times \mathbb B^{n-1}$, it is shown that $M$ is biholomorphic to $Ω$. Further, if the Kobayashi metric on $M$ has corank one, then $M$ is globally a product; in fact, it is biholomorphic to $Z \times \mathbb C$, where $Z \subset Ω= Δ\times \mathbb B^{n-1}$ is a holomorphic retract.

math.CV↗
arXiv2021

Non-negative divisors and the Grauert metric

Grauert showed that it is possible to construct complete Kähler metrics on the complement of complex analytic sets in a domain of holomorphy. In this note, we study the holomorphic sectional curvatures of such metrics on the complement of a principal divisor in $\mathbb{C}^n$, $n \ge 1$. In addition, we also study how this metric and its holomorphic sectional curvature behaves when the corresponding principal divisors vary continuously.

math.CV↗
arXiv2020

Two remarks on the Poincaré metric on a singular Riemann surface foliation

Let $\mathcal{F}$ be a smooth Riemann surface foliation on $M \setminus E$, where $M$ is a complex manifold and $E \subset M$ is a closed set. Fix a hermitian metric $g$ on $M \setminus E$ and assume that all leaves of $\mathcal{F}$ are hyperbolic. For each leaf $L \subset \mathcal{F}$, the ratio of $g | L$, the restriction of $g$ to $L$, and the Poincaré metric $λ_L$ on $L$ defines a positive function $η$ that is known to be continuous on $M \setminus E$ under suitable conditions on $M, E$. For a domain $U \subset M$, we consider $\mathcal{F}_U$, the restriction of $\mathcal{F}$ to $U$ and the corresponding positive function $η_U$ by considering the ratio of $g$ and the Poincaré metric on the leaves of $\mathcal{F}_U$. First, we study the variation of $η_U$ as $U$ varies in the Hausdorff sense motivated by the work of Lins Neto-Martins. Secondly, Minda had shown the existence of a domain Bloch constant for a hyperbolic Riemann surface $S$, which in other words shows that every holomorphic map from the unit disc into $S$, whose distortion at the origin is bounded below, must be locally injective in some hyperbolic ball of uniform radius. We show how to deduce a version of this Bloch constant for $\mathcal{F}$

math.CV↗
arXiv2020

On Grauert's examples of complete Kähler metrics

Grauert showed that the existence of a complete Kähler metric does not characterize domains of holomorphy by constructing such metrics on the complements of complex analytic sets in a domain of holomorphy. In this note, we study the holomorphic sectional curvatures of such metrics in two prototype cases namely, $\mathbb{C}^n \setminus \{0\}, n \ge 2$ and $\mathbb{B}^N \setminus A$, $N \ge 2$ and $A \subset \mathbb{B}^N$ is a hyperplane of codimension at least two. This is done by computing the Gaussian curvature of its restriction to the leaves of a suitable holomorphic foliation of these two examples. We also examine this metric on the punctured plane $\mathbb{C}^{\ast}$ and show that it behaves very differently in this case.

math.CV↗
arXiv2020

Further remarks on the higher dimensional Suita conjecture

For a domain $D \subset \mathbb C^n$, $n \ge 2$, let $F^k_D(z)=K_D(z)λ\big(I^k_D(z)\big)$, where $K_D(z)$ is the Bergman kernel of $D$ along the diagonal and $λ\big(I^k_D(z)\big)$ is the Lebesgue measure of the Kobayashi indicatrix at the point $z$. This biholomorphic invariant was introduced by B\locki and in this note, we study its limiting boundary behaviour on two classes of domains namely, $h$-extendible and strongly pseudoconvex polyhedral 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↗
arXiv2018

On the squeezing function and Fridman invariants

For a domain $D \subset \mathbb C^n$, the relationship between the squeezing function and the Fridman invariants is clarified. Furthermore, localization properties of these functions are obtained. As applications, some known results concerning their boundary behavior are extended.

math.CV↗