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Jaikrishnan Janardhanan

Publications and source records attributed to Jaikrishnan Janardhanan.

14 recordsLinked to original sources
arXiv2026

Holomorphic mappings of finitely connected planar domains

Results on holomorphic mappings of finitely connected planar domains are often proved using Koebe's famous circle mapping theorem. A prominent example is Julia's bound on the size of the automorphism group of finitely connected planar domains of connectivity $k\geq3$ and the sharp bound obtained later by Heins. The purpose of this article is to explore whether it is possible to give a proof of these results, and several other related results, without using Koebe's theorem. The main result is a rigidity theorem for proper holomorphic maps between finitely connected domains of connectivity higher than $2$ that immediately yields a bound on the number of proper holomorphic mappings between two such domains in terms of their connectivities. Using the same ideas, we also provide a new elementary proof of the planar Riemann--Hurwitz formula that does not use the Euler characteristic. Most of our proofs rely only on the Riemann mapping theorem, the Schwarz reflection principle, and the argument principle.

math.CV↗
arXiv2026

Revisiting Kobayashi hyperbolicity on planar domains

We give two new elementary proofs of the complete Kobayashi hyperbolicity of the twice-punctured complex plane. We also present an extremely short proof that bounded domains are complete Kobayashi hyperbolic. Our proofs rely neither on the fact that the universal cover of the twice-punctured plane is the disk nor on the existence of negatively curved metrics. As applications, we present concise proofs of the classical theorems of Landau, Schottky, and Picard. Finally, we provide a characterization of Kobayashi hyperbolicity for planar domains inspired by a similar result of Hahn.

math.CV↗
arXiv2026

Finiteness of the fixed point sets of automorphisms

We investigate the size of fixed point sets of automorphisms of bounded domains in $\mathbb{C}^n$. In one complex variable, a nontrivial automorphism has at most two fixed points, but in higher dimensions fixed point sets need not be discrete. We show, under natural extension hypotheses, that discreteness forces finiteness. We also obtain a uniform bound for the number of fixed points of automorphisms in compact subgroups whose elements admit such extensions.

math.CV↗
arXiv2025

Conformal Mappings Through the Lens of Invariant Metrics

The main objective of this paper is to show that balls under invariant metrics on hyperbolic planar domains are finitely-connected. As applications, we give new and transparent proofs of classical results on conformal mappings of planar domains. In particular, we show that any conformal self-map of a hyperbolic planar domain with three fixed points is the identity. We also give a new and very simple proof of the theorem by Aumann and Carathéodory that states that the isotropy groups of a hyperbolic planar domain are either finite or the domain is simply-connected.

math.CV↗
arXiv2019

Schwarz lemmas via the pluricomplex Green's function

We prove a version of the Schwarz lemma for holomorphic mappings from the unit disk into the symmetric product of a Riemann surface. Our proof is function-theoretic and self-contained. The main novelty in our proof is the use of the pluricomplex Green's function. We also prove several other Schwarz lemmas using this function.

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↗
arXiv2018

A $1$-point Quadrature domain of order $1$ not biholomorphic to a balanced domain

It is known that if $f: D_1 \to D_2$ is a polynomial biholomorphism with polynomial inverse and constant Jacobian then $D_1$ is a $1$-point Quadrature domain (the Bergman span contains all holomorphic polynomials) of order $1$ whenever $D_2$ is a balanced domain. Bell conjectured that all $1$-point Quadrature domains arise in this manner. In this note, we construct a $1$-point Quadrature domain of order $1$ that is not biholomorphic to any balanced domain.

math.CV↗
arXiv2018

A note on the smoothness of the Minkowski function

The Minkowski function is a crucial tool used in the study of balanced domains and, more generally, quasi-balanced domains in several complex variables. If a quasi-balanced domain is bounded and pseudoconvex then it is well-known that its Minkowski function is plurisubharmonic. In this short note, we prove that under the additional assumption of smoothness of the boundary, the Minkowski function of a quasi-balanced domain is in fact smooth away from the origin. This allows us to construct a smooth plurisubharmonic defining function for such domains. Our result is new even in the case of balanced domains.

math.CV↗
arXiv2016

Finiteness Theorems for Products and Symmetric Products of Hyperbolic Riemann Surfaces

We prove that if $X = X_1 \times \dots \times X_n$ is a product of hyperbolic Riemann surfaces of finite type and $Y = Ω/Γ$ is a complex manifold, where $Ω$ is a bounded simply-connected domain in $\mathbb{C}^m$, then the space of dominant holomorphic mappings from $X$ to $Y$ is a finite set. As corollaries, we obtain the finiteness of the space of dominant holomorphic mappings into products and symmetric products of hyperbolic Riemann surfaces.

math.CV↗
arXiv2014

Proper holomorphic mappings of balanced domains in $\mathbb{C}^n$

We extend a well-known result, about the unit ball, by H. Alexander to a class of balanced domains in $\mathbb{C}^n, \ n > 1$. Specifically: we prove that any proper holomorphic self-map of a certain type of balanced, finite-type domain in $\mathbb{C}^n, \ n > 1$, is an automorphism. The main novelty of our proof is the use of a recent result of Opshtein on the behaviour of the iterates of holomorphic self-maps of a certain class of domains. We use Opshtein's theorem, together with the tools made available by finiteness of type, to deduce that the aforementioned map is unbranched. The monodromy theorem then delivers the result.

math.CV↗
arXiv2014

Proper holomorphic maps between bounded symmetric domains revisited

We prove that a proper holomorphic map between two bounded symmetric domains of the same dimension, one of them being irreducible, is a biholomorphism. Our methods allow us to give a single, all-encompassing argument that unifies the various special cases in which this result is known. We discuss an application of these methods to domains having noncompact automorphism groups that are not assumed to act transitively.

math.CV↗
arXiv2013

Proper holomorphic mappings between hyperbolic product manifolds

We prove a result on the structure of finite proper holomorphic mappings between complex manifolds that are products of hyperbolic Riemann surfaces. While an important special case of our result follows from the ideas developed by Remmert and Stein, the proof of the full result relies on the interplay of the latter ideas and a finiteness theorem for Riemann surfaces.

math.CV↗