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G. P. Balakumar

Publications and source records attributed to G. P. Balakumar.

11 recordsLinked to original sources
arXiv2024

Exploring Holomorphic Retracts

The purpose of this article is towards systematically characterizing (holomorphic) retracts of domains of holomorphy; to begin with, bounded balanced pseudoconvex domains $B \subset \mathbb{C}^N$. Specifically, we show that every retract of $B$ passing through its center (origin), is the graph of a holomorphic map over a linear subspace of $B$. To deal with a case when may fail to have sufficiently many extreme points, we consider products of bounded balanced domains of holomorphy with holomorphically extreme boundaries and obtain a complete description of retracts passing through its center. This can be applied to solve a special case of the union problem with a degeneracy, namely: to characterize those Kobayashi corank one complex manifolds which can be expressed as an increasing union of submanifolds which are biholomorphic to a prescribed homogeneous bounded balanced domain. In this course, we prove a generalization of Mazet's Schwarz lemma. Results about non-existence of retracts of each possible dimension is established for the simplest non-convex but pseudoconvex domain: the $\ell^q$-`ball' for all $0<q <1$ as well as `anisotropic' analogues. The same is also done for balanced analytic polyhedra. This enables an illustration of applying retracts to establishing biholomorphic inequivalences. To go beyond balanced domains, we then first obtain a complete characterization of retracts of the Hartogs triangle and `analytic complements' thereof. Thereafter, similar characterization results for domains which are neither bounded nor topologically trivial. We conclude with some expositions about retracts of $\mathbb{C}^2$.

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↗
arXiv2021

Structure theorems for Power Series in Several Complex Variables

It is a classical fact that domains of convergence of power series of several complex variables are characterized as logarithmically convex complete Reinhardt domains; let $D \subsetneq \mathbb{C}^N$ be such a domain. We show that a necessary as well as sufficient condition for a power series $g$ to have $D$ as its domain of convergence is that it admits a certain decomposition into elementary power series; specifically, $g$ can be expressed as a sum of a sequence of power series $g_n$ with the property that each of the logarithmic images $G_n$ of their domains of convergence are half-spaces, all containing the logarithmic image $G$ of $D$ and such that the largest open subset of $\mathbb{C}^N$ on which all the $g_n$'s and $g$ converge absolutely is $D$. In short, every power series admits a decomposition into elementary power series. The proof of this leads to a new way of arriving at a constructive proof of the aforementioned classical fact. This proof inturn leads to another decomposition result in which the $G_n$'s are now wedges formed by intersections of pairs of {\it supporting} half-spaces of $G$. Along the way, we also show that in each fiber of the restriction of the absolute map to the boundary of the domain of convergence of $g$, there exists a singular point of $g$.

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↗
arXiv2016

Power Series in Several Complex Variables

The purpose of this article is to provide an exposition of domains of convergence of power series of several complex variables without recourse to relatively advanced notions of convexity.

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↗
arXiv2011

Some regularity theorems for CR mappings

The purpose of this article is to study Lipschitz CR mappings from an $h$-extendible (or semi-regular) hypersurface in $\mbb C^n$. Under various assumptions on the target hypersurface, it is shown that such mappings must be smooth. A rigidity result for proper holomorphic mappings from strongly pseudoconvex domains is also proved.

math.CV↗