Analysis seminar
Speaker: Kapil Jaglan (IIT Ropar)
Host: Prachi Mahajan
Title: A study on Planar Harmonic Mappings and Minimal surfaces
Time, day and date: 5:00:00 PM, Monday, February 24
Venue: Online (https://meet.google.com/woo-psdw-wdj)
Abstract: This talk explores the properties of univalent harmonic
functions from a geometric function theoretic perspective and
establishes connections between these functions and minimal surfaces.
The talk consists of four sections followed by future direction, with
the initial section serving as an introduction and containing basic
definitions and results from the existing literature.
In the second section, our main aim is to determine a geometric
condition under which a locally univalent harmonic mapping f defined on
the unit disk D is univalent, and maps D onto a linearly accessible
domain of order β for some β ∈ (0, 1). A linearly accessible domain (a
non-convex domain) is important because, under certain sufficient
conditions stated by Dorff et al., minimal graphs over these domains are
area-minimizing. As a consequence, we derive sufficient conditions for f
to map D onto a linearly accessible domain of order β, in the form of a
convolution result, and a coefficient inequality. By extending the ideas
of Dorff et al., we construct one-parameter families of globally
area-minimizing minimal surfaces over a linearly accessible domain of
order β.
In the third section, we explore the properties of odd univalent
harmonic functions. Our starting point of investigation is to obtain the
sharp coefficient estimates for odd univalent functions exhibiting
convexity in one direction. We then advance our investigation to more
generalized classes, including major geometric subclasses of
sense-preserving univalent harmonic mappings. We examine the growth
pattern of odd univalent harmonic functions and extend the range of 'p'
for which these functions belong to the Hardy space h^p. Our results, in
particular, add to the understanding of the growth pattern between odd
univalent harmonic functions and the harmonic Bieberbach conjecture.
In the final section, we deal with the fundamental problem of
determining the location of the zeros of complex-valued harmonic
polynomials. The best-known results available in this direction are up
to harmonic trinomials only. The exploration of the zeros of a general
harmonic polynomial has been limited due to various challenges. Our
research takes a leap further in identifying the regions encompassing
the zeros of a general harmonic polynomial of arbitrary degree using
various techniques, such as the matrix method and certain other matrix
inequalities. Additionally, we employ the harmonic analog of the
argument principle to examine the distribution of zeros, which we
demonstrate through illustrative examples.