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Diganta Borah

Publications and source records attributed to Diganta Borah.

15 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↗
arXiv2024

Some remarks on the Carath\'eodory and Szeg\"o metrics on planar domains

We study several intrinsic properties of the Carath\'eodory and Szeg\"o metrics on finitely connected planar domains. Among them are the existence of closed geodesics and geodesic spirals, boundary behaviour of Gaussian curvatures, and $L^2$-cohomology. A formula for the Szeg\"o metric in terms of the Weierstrass $\wp$-function is obtained. Variations of these metrics and their Gaussian curvatures on planar annuli are also studied. Consequently, we obtain optimal universal upper bounds for their Gaussian curvatures and show that no universal lower bounds exist for their Gaussian curvatures. Moreover, it follows that there are domains where the Gaussian curvature of the Szeg\"o metric assumes both negative and positive values. Lastly, it is also observed that there is no universal upper bound for the ratio of the Szeg\"o and Carath\'eodory metrics.

math.CV↗
arXiv2024

Limits of an increasing sequence of Riemann surfaces

Let $M$ be a Riemann surface which admits an exhaustion by open subsets $M_j$ each of which is biholomorphic to a fixed domain $\Omega \subset \mathbb{C}$. We describe $M$ in terms of $\Omega$ under various assumptions on the boundary components of $\Omega$.

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↗
arXiv2021

Some remarks on the Kobayashi--Fuks metric on strongly pseudoconvex domains

The Ricci curvature of the Bergman metric on a bounded domain $D\subset \mathbb{C}^n$ is strictly bounded above by $n+1$ and consequently $\log (K_D^{n+1}g_{B,D})$, where $K_D$ is the Bergman kernel for $D$ on the diagonal and $g_{B, D}$ is the Riemannian volume element of the Bergman metric on $D$, is the potential for a K\"ahler metric on $D$ known as the Kobayashi--Fuks metric. In this note we study the localization of this metric near holomorphic peak points and also show that this metric shares several properties with the Bergman metric 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 $\Omega \subset \mathbb C^n$. The main question addressed here is to describe $M$ in terms of $\Omega$. 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 $\Omega$ 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 $\Omega$ 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 $\Omega$ 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 $\Omega$ or that of a limiting domain associated with it. Finally, when $\Omega = \Delta \times \mathbb B^{n-1}$, the product of the unit disc $\Delta \subset \mathbb C$ 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 $\Omega = \Delta \times \mathbb B^{n-1}$, it is shown that $M$ is biholomorphic to $\Omega$. 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 $Z \subset \Omega = \Delta \times \mathbb B^{n-1}$ is a holomorphic retract.

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 K\"{a}hler 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↗
arXiv2018

Further remarks on the higher dimensional Suita conjecture

For a domain $D \subset \mathbb C^n$, $n \ge 2$, let $F^k_D(z)=K_D(z)\lambda\big(I^k_D(z)\big)$, where $K_D(z)$ is the Bergman kernel of $D$ along the diagonal and $\lambda\big(I^k_D(z)\big)$ 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↗
arXiv2018

Remarks on the higher dimensional Suita conjecture

To study the analog of Suita's conjecture for domains $D \subset \mathbb{C}^n$, $n \ge 2$, B\l ocki introduced the invariant $F^k_D(z)=K_D(z)\lambda\big(I^k_D(z)\big)$, where $K_D(z)$ is the Bergman kernel of $D$ along the diagonal and $\lambda\big(I^k_D(z)\big)$ is the Lebesgue measure of the Kobayashi indicatrix at the point $z$. In this note, we study the behaviour of $F^k_D(z)$ (and other similar invariants using different metrics) on strongly pseudconvex domains and also compute its limiting behaviour explicitly at certain points of decoupled egg domains in $\mathbb{C}^2$.

math.CV↗
arXiv2017

Arakelov Self-intersection numbers of minimal regular models of modular curves $X_0(p^2)$

We compute an asymptotic expression for the Arakelov self-intersection number of the relative dualizing sheaf of Edixhoven's minimal regular model for the modular curve $X_0(p^2)$ over $\mathbb{Q}$. The computation of the self-intersection numbers are used to prove effective Bogolomov conjecture for the semi-stable models of modular curves $X_0(p^2)$ and obtain a bound on the stable Faltings height for those curves in a companion article arXiv:1802.06968.

math.NT↗
arXiv2016

Comments on the Green's function of a planar domain

We study several quantities associated to the Green's function of a multiply connected domain in the complex plane. Among them are some intrinsic properties such as geodesics, curvature, and $L^2$-cohomology of the capacity metric and critical points of the Green's function. The principal idea used is an affine scaling of the domain that furnishes quantitative boundary behaviour of the Green's function and related objects.

math.CV↗
arXiv2014

Remarks on the metric induced by the Robin function III

Let $D$ be a smoothly bounded pseudoconvex domain in $\mathbf C^n$, $n > 1$. Using the Robin function $\Lambda(p)$ that arises from the Green function $G(z, p)$ for $D$ with pole at $p \in D$ associated with the standard sum-of-squares Laplacian, N. Levenberg and H. Yamaguchi had constructed a K\"{a}hler metric (the so-called $\Lambda$-metric) on $D$. In this article, we study the existence of geodesic spirals for this metric.

math.CV↗
arXiv2012

Remarks on the metric induced by the Robin function II

Let $D$ be a smoothly bounded pseudoconvex domain in $\mathbf C^n$, $n > 1$. Using the Robin function $\La(p)$ that arises from the Green function $G(z, p)$ for $D$ with pole at $p \in D$ associated with the standard sum-of-squares Laplacian, N. Levenberg and H. Yamaguchi had constructed a K\"{a}hler metric (the so-called $\La$-metric) on $D$. Assume that $D$ is strongly pseudoconvex and $ds^2$ denotes the $\La$-metric on $D$. In this article, first we prove that the holomorphic sectional curvature of $ds^2$ along normal directions converges to a negative constant near the boundary of $D$. Then, we prove that if $D$ is not simply connected, then any nontrivial homotopy class of $\pi_1(D)$ contains a closed geodesic for $ds^2$. Finally, we prove that the diminesion of the space of square integrable harmonic $(p, q)$-forms on $D$ relative to $ds^2$ is zero except when $p+q=n$ in which case it is infinite.

math.CV↗
arXiv2010

Remarks on the metric induced by the Robin function

Let $D$ be a smoothly bounded pseudoconvex domain in $\mathbf C^n$, $n > 1$. Using $G(z, p)$, the Green function for $D$ with pole at $p \in D$ associated with the standard sum-of-squares Laplacian, N. Levenberg and H. Yamaguchi had constructed a Kähler metric (the so-called $\La$-metric) using the Robin function $\La(p)$ arising from $G(z, p)$. The purpose of this article is to study this metric by deriving its boundary asymptotics and using them to calculate the holomorphic sectional curvature along normal directions. It is also shown that the $\La$-metric is comparable to the Kobayashi (and hence to the Bergman and Carathéodory metrics) when $D$ is strongly pseudoconvex. The unit ball in $\mathbf C^n$ is also characterized among all smoothly bounded strongly convex domains on which the $\La$-metric has constant negative holomorphic sectional curvature. This may be regarded as a version of Lu-Qi Keng's theorem for the Bergman metric.

math.CV↗