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On the Uniqueness of Vortex Equations and Its Geometric Applications

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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 \(\partial u\ne 0\) 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.

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References

  1. Benoist, Y., Hulin, D.: Cubic differentials and hyperbolic convex sets. J. Differ. Geom. 98(1), 1–19 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Calabi, E.: Complete affine hyperspheres. I. In: Symposia Mathematica, Vol. X, pp. 19–38 (Convegno di Geometria Differenziale, INDAM, Rome, 1971). Academic Press, London (1972)

  3. Cheng, S.Y., Yau, S.T.: Differential equations on Riemannian manifolds and their geometric applications. Commun. Pure Appl. Math. 28(3), 333–354 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  4. Choi, H.I., Treibergs, A.: Gauss maps of spacelike constant mean curvature hypersurfaces of Minkowski space. J. Differ. Geom. 32(3), 775–817 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dai, S., Li, Q.: Minimal surfaces for hitchin representations. J. Differ. Geom. (to appear )

  6. Dumas, D., Wolf, M.: Polynomial cubic differentials and convex polygons in the projective plane. Geom. Funct. Anal. 25(6), 1734–1798 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Han, Z.-C.: Remarks on the geometric behavior of harmonic maps between surfaces. In: Elliptic and Parabolic Methods in Geometry (Minneapolis, MN, 1994), pp. 57–66. A K Peters, Wellesley, MA (1996)

  8. Han, Z.-C., Tam, L.-F., Treibergs, A., Wan, T.: Harmonic maps from the complex plane into surfaces with nonpositive curvature. Commun. Anal. Geom. 3(1–2), 85–114 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  9. Li, A.M.: Calabi conjecture on hyperbolic affine hyperspheres. Math. Z. 203(3), 483–491 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  10. Li, A.M.: Calabi conjecture on hyperbolic affine hyperspheres. II. Math. Ann. 293(3), 485–493 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  11. Li, A.-M., Li, H., Simon, U.: Centroaffine Bernstein problems. Differ. Geom. Appl. 20(3), 331–356 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Milnor, T.K.: Harmonic maps and classical surface theory in Minkowski \(3\)-space. Trans. Am. Math. Soc. 280(1), 161–185 (1983)

    MathSciNet  MATH  Google Scholar 

  13. Omori, H.: Isometric immersions of Riemannian manifolds. J. Math. Soc. Jpn. 19, 205–214 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  14. Osserman, R.: A Survey of Minimal Surfaces. Courier Corporation, North Chelmsford (2002)

    MATH  Google Scholar 

  15. Sampson, J.H.: Some properties and applications of harmonic mappings. Ann. Sci. École Norm. Sup. (4) 11(2), 211–228 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  16. Schoen, R., Yau, S.T.: On univalent harmonic maps between surfaces. Invent. Math. 44(3), 265–278 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  17. Simon, U., Wang, C.P.: Local theory of affine 2-spheres. In: Differential Geometry: Riemannian Geometry (Los Angeles, CA, 1990), Proceedings of Symposia in Pure Mathematics, vol. 54, pp. 585–598. American Mathematical Society, Providence, RI (1993)

  18. Wan, T.Y.-H.: Constant mean curvature surface, harmonic maps, and universal Teichmüller space. J. Differ. Geom. 35(3), 643–657 (1992)

    Article  MATH  Google Scholar 

  19. Wan, T.Y.H., Au, T.K.-K.: Parabolic constant mean curvature spacelike surfaces. Proc. Am. Math. Soc. 120(2), 559–564 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  20. Wang, C.: Some examples of complete hyperbolic affine 2-spheres in \({\bf R}^{3}\). In: Global Differential Geometry and Global Analysis (Berlin, 1990). Lecture Notes in Mathematics, vol. 1481, Springer, Berlin, pp. 271–280 (1991)

  21. Yau, S.T.: Harmonic functions on complete Riemannian manifolds. Commun. Pure Appl. Math. 28, 201–228 (1975)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

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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  1. Centre for Quantum Geometry of Moduli Spaces (QGM), Aarhus University, Ny Munkegade 118, 8000, Aarhus C, Denmark

    Qiongling Li

  2. Department of Mathematics, California Institute of Technology, 1200 East California Boulevard, Pasadena, USA

    Qiongling Li

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Li, Q. On the Uniqueness of Vortex Equations and Its Geometric Applications. J Geom Anal 29, 105–120 (2019). https://doi.org/10.1007/s12220-018-9981-x

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  • DOI: https://doi.org/10.1007/s12220-018-9981-x

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