Indian Institute of Technology Palakkad&nbsp;(IIT Palakkad or IIT PKD) is a public autonomous&nbsp;engineering&nbsp;and&nbsp;research&nbsp;institute located in&nbsp;Palakkad,&nbsp;Kerala. It was&nbsp;one of the five new&nbsp;IITs&nbsp;proposed in the&nbsp;2014 Union budget of India. The campus was inaugurated on 3 August 2015. The institute has 94 faculty, 978 students, and 63 non-teaching staffs.

IT Palakkad currently offers four-year Bachelor of Technology (B.Tech) and two years MS programmes in 8 different engineering disciplines including&nbsp;Computer Science, Electrical, Mechanical, Civil among others.

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IIT Palakkad

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. © 2020 Informa UK Limited, trading as Taylor & Francis Group.

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Department of Mathematics

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Complex Variables and Elliptic Equations

A note on the smoothness of the Minkowski 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. © 2019, Mathematica Josephina, Inc.

Journal of Geometric Analysis

Schwarz Lemmas via the Pluricomplex Green’s Function

It is known that if f: D1→ D2 is a polynomial biholomorphism with polynomial inverse and constant Jacobian then D1 is a 1-point poly-quadrature domain (the Bergman span contains all holomorphic polynomials) of order 1 whenever D2 is a complete circular domain. Bell conjectured that all 1-point poly-quadrature domains arise in this manner. In this note, we construct a 1-point poly-quadrature domain of order 1 that is not biholomorphic to any complete circular domain. © 2018, Springer Nature Switzerland AG.

Analysis and Mathematical Physics

A 1-point poly-quadrature domain of order 1 not biholomorphic to a complete circular domain

arXiv preprint arXiv:1807.

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

We show that the structure of proper holomorphic maps between the n-fold symmetric products, n ≥ 2, of a pair of noncompact 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 noncompact Riemann surface. © 2018 Deutsche Mathematiker Vereinigung.

Documenta Mathematica

Proper holomorphic mappings onto symmetric products of a Riemann surface

We prove that the space of dominant/non-constant holomorphic mappings from a product of hyperbolic Riemann surfaces of finite type into certain hyperbolic manifolds that can be covered by a bounded domain is a finite set. © 2017 World Scientific Publishing Company.

International Journal of Mathematics

Finiteness theorems for holomorphic mapping from products of hyperbolic Riemann surfaces

We extend a well-known result, about the unit ball, by H. Alexander to a class of balanced domains in (Formula presented.),n>1. Specifically: we prove that any proper holomorphic self-map of a certain type of balanced, finite-type domain in (Formula presented.),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. © 2015, Springer-Verlag Berlin Heidelberg.

Mathematische Zeitschrift

Proper holomorphic mappings of balanced domains in Cn

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. © 2014 World Scientific Publishing Company.

Proper holomorphic mappings between hyperbolic product manifolds

We prove that a proper holomorphic map between two nonplanar 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. © 2014 Mathematical Sciences Publishers.