The Journal of Geometric Analysis

, Volume 29, Issue 1, pp 105–120 | Cite as

On the Uniqueness of Vortex Equations and Its Geometric Applications

  • Qiongling LiEmail author


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 \(\partial u\ne 0\) 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 finitely many zeros.


Vortex equations Polynomial differentials Harmonic maps 

Mathematics Subject Classification

53C21 53C43 58E20 



The author wishes to thank Vlad Markovic, Song Dai and Mike Wolf for helpful discussions. The author is supported by the center of excellence grant ‘Center for Quantum Geometry of Moduli Spaces’ from the Danish National Research Foundation (DNRF95). She also acknowledges the support from U.S. National Science Foundation Grants DMS 1107452, 1107263, 1107367 “RNMS: GEometric structures And Representation varieties” (the GEAR Network).


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Authors and Affiliations

  1. 1.Centre for Quantum Geometry of Moduli Spaces (QGM)Aarhus UniversityAarhus CDenmark
  2. 2.Department of MathematicsCalifornia Institute of TechnologyPasadenaUSA

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