arXiv subjects

Prachi Mahajan

Publications and source records attributed to Prachi Mahajan.

12 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↗

arXiv2025

A Hermitian metric on hyperbolic complex manifolds

We describe a method of defining a Hermitian metric on Kobayashi hyperbolic manifolds. The metric is distance decreasing under holomorphic mappings, up to a multiplicative constant. This method is distinct from the classical construction of Wu, and yields a metric which is expected to have superior regularity properties.

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 $Ω\subset \mathbb{C}$. We describe $M$ in terms of $Ω$ under various assumptions on the boundary components of $Ω$.

math.CV↗

arXiv2024

The Bergman-Fridman invariant on some classes of pseudoconvex domains

We study the boundary behaviour of a variant of the Fridman's invariant function (defined in terms of the Bergman metric) on Levi corank one domains, strongly pseudoconvex domains, smoothly bounded convex domains in $ \mathbb{C}^n $ and polyhedral domains in $ \mathbb{C}^2 $.

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↗

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↗

arXiv2018

Remarks on the higher dimensional Suita conjecture

To study the analog of Suita's conjecture for domains $D \subset \mathbb{C}^n$, $n \ge 2$, Błocki introduced the invariant $F^k_D(z)=K_D(z)λ\big(I^k_D(z)\big)$, 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 $z$. 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↗

arXiv2018

A comparison of two biholomorphic invariants

The Fridman invariant, which is a biholomorphic invariant on Kobayashi hyperbolic manifolds, can be seen as the dual of the much studied squeezing function. We compare this pair of invariants by showing that they are both equally capable of determining the boundary geometry of a bounded domain if their boundary behaviour is apriori known.

math.CV↗

arXiv2015

Analyzing the Wu metric on a class of eggs in $\mathbb{C}^n$ -- I

We study the Wu metric on convex egg domains of the form \[ E_{2m} = \big\{ z \in \mathbb{C}^n : \vert z_1 \vert^{2m} + \vert z_2 \vert^2 + \ldots + \vert z_{n-1} \vert^2 + \vert z_n \vert^{2} <1 \big\} \] where $m \geq 1/2, m \neq 1$. The Wu metric is shown to be real analytic everywhere except on a lower dimensional subvariety where it fails to be $C^2$-smooth. Overall however, the Wu metric is shown to be continuous when $m=1/2$ and even $C^1$-smooth for each $m>1/2$, and in all cases, a non-Kähler Hermitian metric with its holomorphic curvature strongly negative in the sense of currents. This gives a natural answer to a conjecture of S. Kobayashi and H. Wu for such $E_{2m}$.

math.CV↗

arXiv2015

Analyzing the Wu metric on a class of eggs in $\mathbb{C}^n$ -- II

We study the Wu metric for the non-convex domains of the form \[ E_{2m} = \big\{ z \in \mathbb{C}^n : \vert z_1 \vert^{2m} + \vert z_2 \vert^2 + \ldots + \vert z_{n-1} \vert^2 + \vert z_n \vert^{2} <1 \big \}, \] where $ 0 < m < 1/2$. Explicit expressions for the Kobayashi metric and the Wu metric on such pseudo-eggs $E_{2m}$ are obtained. The Wu metric is then verified to be a continuous Hermitian metric on $ E_{2m} $ which is real analytic everywhere except along the complex hypersurface $ Z = \{ (0, z_2, \ldots, z_n ) \in E_{2m} \} $. We also show that the holomorphic sectional curvature of the Wu metric for this non-compact family of pseudoconvex domains is bounded above in the sense of currents by a negative constant independent of $m$. This verifies a conjecture of S. Kobayashi and H. Wu for such $E_{2m}$.

math.CV↗

arXiv2013

Bounds for Invariant Distances on Pseudoconvex Levi Corank One Domains and Applications

Let $D \subset \mathbb{C}^n$ be a smoothly bounded pseudoconvex Levi corank one domain with defining function $r$, i.e., the Levi form $\partial \bar {\partial} r$ of the boundary $\partial D$ has at least $(n - 2)$ positive eigenvalues everywhere on $\partial D$. The main goal of this article is to obtain a lower bound for the Carathéodory, Kobayashi and the Bergman distance between a given pair of points $p, q \in D$ in terms of parameters that reflect the Levi geometry of $\partial D$ and the distance of these points to the boundary. Applications include an understanding of Fridman's invariant for the Kobayashi metric on Levi corank one domains, a description of the balls in the Kobayashi metric on such domains that are centered at points close to the boundary in terms of Euclidean data and the boundary behaviour of Kobayashi isometries from such domains.

math.CV↗

arXiv2010

On isometries of the Kobayashi and Carathéodory metrics

This article considers isometries of the Kobayashi and Carathéod-ory metrics on domains in $ \mathbf{C}^n $ and the extent to which they behave like holomorphic mappings. First we prove a metric version of Poincaré's theorem about biholomorphic inequivalence of $ \mathbf{B}^n $, the unit ball in $ \mathbf{C}^n $ and $ Δ^n $, the unit polydisc in $ \mathbf{C}^n $ and then provide few examples which \textit{suggest} that $ \mathbf{B}^n $ cannot be mapped isometrically onto a product domain. In addition, we prove several results on continuous extension of isometries $ f : D_1 \rightarrow D_2 $ to the closures under purely local assumptions on the boundaries. As an application, we show that there is no isometry between a strongly pseudoconvex domain in $ \mathbf{C}^2 $ and certain classes of weakly pseudoconvex finite type domains in $ \mathbf{C}^2 $.

math.CV↗