On the uniqueness of vortex equations and its geometric applications

TL;DR

This paper investigates the uniqueness of solutions to a vortex equation involving entire functions, demonstrating unique harmonic maps and affine spherical immersions with polynomial differentials, while showing non-uniqueness for non-polynomial cases.

Contribution

It establishes the uniqueness of certain vortex equations with polynomial differentials and explores geometric applications, including harmonic maps and affine spherical immersions.

Findings

Unique harmonic maps with polynomial Hopf differential

Unique affine spherical immersions with polynomial Pick differential

Non-uniqueness for non-polynomial entire functions with finite zeros

Abstract

We study the uniqueness of a vortex equation involving an entire function on the complex plane. As geometric applications, we show that there is a unique harmonic map $u:\mathbb{C}\rightarrow \mathbb{H}^2$ satisfying $\partial u\neq 0$ with prescribed polynomial Hopf differential; there is a unique affine spherical immersion $u:\mathbb{C}\rightarrow \mathbb{R}^3$ with prescribed polynomial Pick differential. We also show that the uniqueness fails for non-polynomial entire functions with finite zeros.