On the uniqueness of vortex equations and its geometric applications

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 $u:\mathbb{C}\to {\mathbb{H}}^2$ satisfying $\partial u\neq 0$ with prescribed polynomial Hopf differential; there is a unique affine spherical immersion $u:\mathbb{C}\to {\mathbb{R}}^3$ with prescribed polynomial Pick differential. We also show that the uniqueness fails for non-polynomial entire functions with finite zeros.