A study on planar harmonic mappings and minimal surfaces

Abstract

It is well-known that minimal surfaces over convex domains are always globally area-minimizing, which is not necessarily true for minimal surfaces over non-convex domains. Recently, M. Dorff, D. Halverson, and G. Lawlor proved that minimal surfaces over a bounded linearly accessible domain D of order β for some β ∈ (0, 1) must be globally area-minimizing, provided a certain geometric inequality is satisfied on the boundary of D. We prove sufficient conditions for a sense-preserving harmonic function f = h+g to be linearly accessible of order β. Then, we provide a method to construct harmonic polynomials which map the open unit disk |z| < 1 onto a linearly accessible domain of order β. Using these harmonic polynomials, we construct one parameter families of globally area-minimizing minimal surfaces over non-convex domains. We explore odd univalent harmonic mappings, focusing on coefficient estimates, growth and distortion theorems. Odd univalent analytic functions played an instrumental role in the proof of the celebrated Bieberbach conjecture. Motivated by the unresolved harmonic analog of the Bieberbach conjecture, we investigate specific subclasses of odd functions in S0H , the class of sense-preserving univalent harmonic functions. We provide sharp coefficient bounds for odd univalent harmonic functions exhibiting convexity in one direction and extend our findings to a more generalized class, including the major geometric subclasses of odd functions in S0H . Additionally, we analyze the inclusion of these functions in Hardy spaces and broaden the range of p for which they belong. In particular, the results enhance understanding and highlight analogous growth patterns between odd univalent harmonic functions and the harmonic Bieberbach conjecture. We also propose two conjectures and possible scope for further study as well. We prove sufficient conditions for a normalized complex-valued harmonic function f defined on the unit disk to be univalent and convex in one direction/close-to-convex. Using the geometric properties of convex in one direction or close-to-convex function, we obtain sufficient conditions for univalency in terms of certain integral inequalities. With the help of an integral inequality, we prove a sharp coefficient criterion for f to be convex in one direction. As an application, we finally generate families of univalent harmonic mappings convex in one direction using Gaussian hypergeometric functions. Lastly, our attention is directed towards the zeros of the harmonic polynomials. In their groundbreaking work, Khavinson and ´Swi¸atek proved Wilmshurst’s conjecture, establishing a sharp upper bound on the number of zeros of harmonic polynomials of the form h(z) − z, where h(z) is an analytic polynomial of degree greater than one. Recent studies by Dorff et al. and Liu et al. further determined the number of zeros and the compact region containing all zeros of harmonic trinomials, respectively. Our research takes a leap further in identifying the precise compact region encompassing all zeros of general harmonic polynomials. Moreover, we utilize the harmonic analog of the argument principle to explore the distribution of zeros of these polynomials, offering insightful examples for clarification.