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Gautam Bharali

Publications and source records attributed to Gautam Bharali.

At least 19 recordsLinked to original sources
arXiv2026

Rarity of $\boldsymbol{\mathcal{C}^{1,1}}$ solutions to the complex Monge--Ampère equation on weakly pseudoconvex domains

We show that on any weakly pseudoconvex $B$-regular domain, the classical Dirichlet problem for the complex Monge--Ampère equation with $\mathcal{C}^\infty$-smooth data does not in general admit $\mathcal{C}^{1,1}$-smooth solutions. This working draft is a prelude to potential-theoretic solutions to some extension problems for mappings that were thought to rely on such $\mathcal{C}^{1,1}$-smooth solutions.

math.CV↗
arXiv2025

On some connections between Kobayashi geometry and pluripotential theory

In this paper, we explore some connections between Kobayashi geometry and the Dirichlet problem for the complex Monge--Ampère equation. Among the results we obtain through these connections are: $(i)$~a theorem on the continuous extension up to $\partial{D}$ of a proper holomorphic map $F: D\longrightarrow Ω$ between domains with $\dim_{\mathbb{C}}(D) < \dim_{\mathbb{C}}(Ω)$, and $(ii)$~a result that establishes the existence of bounded domains with ``nice'' boundary geometry on which Hölder regularity of the solutions to the complex Monge--Ampère equation fails. The first, a result in Kobayashi geometry, relies upon an auxiliary construction that involves solving the complex Monge--Ampère equation with Hölder estimates. The second result relies crucially on a bound for the Kobayashi metric.

math.CV↗
arXiv2025

Unbounded visibility domains: metric estimates and an application

We give an explicit lower bound, in terms of the distance from the boundary, for the Kobayashi metric of a certain class of bounded pseudoconvex domains in $\mathbb{C}^n$ with $\mathcal{C}^2$-smooth boundary using the regularity theory for the complex Monge--Ampere equation. Using such an estimate, among other tools, we construct a family of unbounded Kobayashi hyperbolic domains in $\mathbb{C}^n$ having a certain negative-curvature-type property with respect to the Kobayashi distance. As an application, we prove a Picard-type extension theorem for the latter domains.

math.CV↗
arXiv2024

Explicit universal bounds for squeezing functions of ($\mathbb{C}$-)convex domains

We prove two separate lower bounds -- one for nondegenerate convex domains and the other for nondegenerate $\mathbb{C}$-convex (but not necessarily convex) domains -- for the squeezing function that hold true for all domains in $\mathbb{C}^n$, for a fixed $n\geq 2$, of the stated class. We provide explicit expressions in terms of $n$ for these estimates.

math.CV↗
arXiv2023

The squeezing function: exact computations, optimal estimates, and a new application

We present a new application of the squeezing function $s_D$, using which one may detect when a given bounded pseudoconvex domain $D\varsubsetneq \mathbb{C}^n$, $n\geq 2$, is not biholomorphic to any product domain. One of the ingredients used in establishing this result is also used to give an exact computation of the squeezing function (which is a constant) of any bounded symmetric domain. This extends a computation by Kubota to any Cartesian product of Cartan domains at least one of which is an exceptional domain. Our method circumvents any case-by-case analysis by rank and also provides optimal estimates for the squeezing functions of certain domains. Lastly, we identify a family of bounded domains that are holomorphic homogeneous regular.

math.CV↗
arXiv2023

Unbounded visibility domains, the end compactification, and applications

In this paper we study when the Kobayashi distance on a Kobayashi hyperbolic domain has certain visibility properties, with a focus on unbounded domains. "Visibility" in this context is reminiscent of visibility, seen in negatively curved Riemannian manifolds, in the sense of Eberlein-O'Neill. However, we do not assume that the domains studied are Cauchy-complete with respect to the Kobayashi distance, as this is hard to establish for domains in $\mathbb{C}^d$, $d \geq 2$. We study the various ways in which this property controls the boundary behavior of holomorphic maps. Among these results is a Carathéodory-type extension theorem for biholomorphisms between planar domains, notably: between infinitely-connected domains. We also explore connections between our visibility property and Gromov hyperbolicity of the Kobayashi distance.

math.CV↗
arXiv2023

A new family of holomorphic homogeneous regular domains and some questions on the squeezing function

We revisit the phenomenon where, for certain domains $D$, if the squeezing function $s_D$ extends continuously to a point $p\in \partial{D}$ with value $1$, then $\partial{D}$ is strongly pseudoconvex around $p$. In $\mathbb{C}^2$, we present weaker conditions under which the latter conclusion is obtained. In another direction, we show that there are bounded domains $D\Subset \mathbb{C}^n$, $n\geq 2$, that admit large $\partial{D}$-open subsets $\mathscr{O}\subset \partial{D}$ such that $s_D\to 0$ approaching any point in $\mathscr{O}$. This is impossible for planar domains. We pose a few questions related to these phenomena. But the core result of this paper identifies a new family of holomorphic homogeneous regular domains. We show via a family of examples how abundant domains satisfying the conditions of this result are.

math.CV↗
arXiv2021

A weak notion of visibility, a family of examples, and Wolff--Denjoy theorems

We investigate a form of visibility introduced recently by Bharali and Zimmer -- and shown to be possessed by a class of domains called Goldilocks domains. The range of theorems established for these domains stem from this form of visibility together with certain quantitative estimates that define Goldilocks domains. We show that some of the theorems alluded to follow merely from the latter notion of visibility. We call those domains that possess this property visibility domains with respect to the Kobayashi distance. We provide a sufficient condition for a domain in $\mathbb{C}^n$ to be a visibility domain. A part of this paper is devoted to constructing a family of domains that are visibility domains with respect to the Kobayashi distance but are not Goldilocks domains. Our notion of visibility is reminiscent of uniform visibility in the context of CAT(0) spaces. However, this is an imperfect analogy because, given a bounded domain $Ω$ in $\mathbb{C}^n$, $n\geq 2$, it is, in general, not even known whether the metric space $(Ω,{\sf k}_Ω)$ (where ${\sf k}_Ω$ is the Kobayashi distance) is a geodesic space. Yet, with just this weak property, we establish two Wolff--Denjoy-type theorems.

math.CV↗
arXiv2021

The entropy of holomorphic correspondences: exact computations and rational semigroups

We study two notions of topological entropy of correspondences introduced by Friedland and Dinh-Sibony. Upper bounds are known for both. We identify a class of holomorphic correspondences whose entropy in the sense of Dinh-Sibony equals the known upper bound. This provides an exact computation of the entropy for rational semigroups. We also explore a connection between these two notions of entropy.

math.DS↗
arXiv2021

On the growth of the Bergman metric near a point of infinite type

We derive optimal estimates for the Bergman kernel and the Bergman metric for certain model domains in $\mathbb{C}^2$ near boundary points that are of infinite type. Being unbounded models, these domains obey certain geometric constraints -- some of them necessary for a non-trivial Bergman space. However, these are mild constraints: unlike most earlier works on this subject, we are able to make estimates for non-convex pseudoconvex models as well. In fact, the domains we can analyse range from being mildly infinite-type to very flat at infinite-type boundary points.

math.CV↗
arXiv2018

Proper holomorphic mappings onto symmetric products of a Riemann surface

We show that the structure of proper holomorphic maps between the $n$-fold symmetric products, $n\geq 2$, of a pair of non-compact Riemann surfaces $X$ and $Y$, provided these are reasonably nice, is very rigid. Specifically, any such map is determined by a proper holomorphic map of $X$ onto $Y$. This extends existing results concerning bounded planar domains, and is a non-compact analogue of a phenomenon observed in symmetric products of compact Riemann surfaces. Along the way, we also provide a condition for the complete hyperbolicity of all $n$-fold symmetric products of a non-compact Riemann surface.

math.CV↗
arXiv2017

Holomorphic correspondences related to finitely generated rational semigroups

In this paper, we present a new technique for studying the dynamics of a finitely generated rational semigroup. Such a semigroup can be associated naturally to a certain holomorphic correspondence on $\mathbb{P}^1$. Then, results on the iterative dynamics of such a correspondence can be applied to the study of the rational semigroup. We focus on a certain invariant measure for the aforementioned correspondence---known as the equilibrium measure. This confers some advantages over many of the known techniques for studying the dynamics of rational semigroups. We use the equilibrium measure to analyse the distribution of repelling fixed points of a finitely generated rational semigroup, and to derive a sharp bound for the Hausdorff dimension of the Julia set of such a semigroup.

math.DS↗
arXiv2017

Goldilocks domains, a weak notion of visibility, and applications

In this paper we introduce a new class of domains in complex Euclidean space, called Goldilocks domains, and study their complex geometry. These domains are defined in terms of a lower bound on how fast the Kobayashi metric grows and an upper bound on how fast the Kobayashi distance grows as one approaches the boundary. Strongly pseudoconvex domains and weakly pseudoconvex domains of finite type always satisfy this Goldilocks condition, but we also present families of Goldilocks domains that have low boundary regularity or have boundary points of infinite type. We will show that the Kobayashi metric on these domains behaves, in some sense, like a negatively curved Riemannian metric. In particular, it satisfies a visibility condition in the sense of Eberlein and O'Neill. This behavior allows us to prove a variety of results concerning boundary extension of maps and to establish Wolff-Denjoy theorems for a wide collection of domains.

math.CV↗
arXiv2016

A criterion for a degree-one holomorphic map to be a biholomorphism

Let $X$ and $Y$ be compact connected complex manifolds of the same dimension with $b_2(X)= b_2(Y)$. We prove that any surjective holomorphic map of degree one from $X$ to $Y$ is a biholomorphism. A version of this was established by the first two authors, but under an extra assumption that $\dim H^1(X {\mathcal O}_X)\,=\,\dim H^1(Y {\mathcal O}_Y)$. We show that this condition is actually automatically satisfied.

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 dynamics of holomorphic correspondences of P^1: invariant measures and the normality set

This paper is motivated by Brolin's theorem. The phenomenon we wish to demonstrate is as follows: if $F$ is a holomorphic correspondence on $\mathbb{P}^1$, then (under certain conditions) $F$ admits a measure $μ_F$ such that, for any point $z$ drawn from a "large" open subset of $\mathbb{P}^1$, $μ_F$ is the weak*-limit of the normalised sums of point masses carried by the pre-images of $z$ under the iterates of $F$. Let ${}^\dagger{F}$ denote the transpose of $F$. Under the condition $d_{top}(F) > d_{top}({}^\dagger{F})$, where $d_{top}$ denotes the topological degree, the above phenomemon was established by Dinh and Sibony. We show that the support of this $μ_F$ is disjoint from the normality set of $F$. There are many interesting correspondences on $\mathbb{P}^1$ for which $d_{top}(F) \leq d_{top}({}^\dagger{F})$. Examples are the correspondences introduced by Bullett and collaborators. When $d_{top}(F) \leq d_{top}({}^\dagger{F})$, equidistribution cannot be expected to the full extent of Brolin's theorem. However, we prove that when $F$ admits a repeller, equidistribution in the above sense holds true.

math.CV↗
arXiv2015

Complex geodesics, their boundary regularity, and a Hardy--Littlewood-type lemma

We begin by giving an example of a smoothly bounded convex domain that has complex geodesics that do not extend continuously up to $\partial\mathbb{D}$. This example suggests that continuity at the boundary of the complex geodesics of a convex domain $Ω\Subset \mathbb{C}^n$, $n\geq 2$, is affected by the extent to which $\partialΩ$ curves or bends at each boundary point. We provide a sufficient condition to this effect (on $\mathcal{C}^1$-smoothly bounded convex domains), which admits domains having boundary points at which the boundary is infinitely flat. Along the way, we establish a Hardy--Littlewood-type lemma that might be of independent interest.

math.CV↗