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On the Existence of Non-CSC Extremal Kähler Metrics with Finite Singularities on \(S^2\)

  • Zhiqiang WeiEmail author
  • Yingyi Wu
Article
  • 25 Downloads

Abstract

We often call an extremal Kähler metric with finite singularities on a compact Riemann surface an HCMU (the Hessian of the Curvature of the Metric is Umbilical) metric. In this paper, we consider the problem: suppose \(\alpha _{1},\alpha _{2},\ldots ,\alpha _{N}\) are \(N\ge 4\) nonnegative real numbers with \(\alpha _{j}\ge 2\;(1\le j\le J\le N-3)\) being integers such that
$$\begin{aligned} \sum _{j=1}^{J}\alpha _{j}+2-N\ge 0, \end{aligned}$$
given any J points \(p_{1},\ldots ,p_{J}\) on \(S^{2}\setminus \{0,\infty \}\), whether there exists a non-CSC conformal HCMU metric g with singular angles\(2\pi \alpha _{1},\ldots ,2\pi \alpha _{N}\), which belongs to the first class (see Definition 1.1) such that \(p_{1},\ldots ,p_{J}\) are all saddle points of scalar curvature R of g and \(0,\infty \) are extremal point of R. We will give a sufficient condition when R has only one saddle point. As its application, we prove that when the number of the singularities is 4, Obstruction Theorem is also a sufficient condition for the existence of a non-CSC conformal HCMU metric on \(S^{2}\).

Keywords

Extremal Kähler metric Conical singularities Cusp singularities 

Mathematics Subject Classification

53C21 53C56 

Notes

Acknowledgements

Yingyi Wu is supported by NSFC No. 11471308. The authors would like to express their deep gratitude to the referee for his/her very valuable comments on improving the whole paper. This work is also supported by the National Natural Science Foundation of China (Grant No. 11871450).

References

  1. 1.
    Aubin, T.: Nonlinear Analysis on Manifolds, Monge Ampre Equations. Grundlehren der Mathematicschen Wissenchaften, vol. 252. Springer, New York (1982)CrossRefGoogle Scholar
  2. 2.
    Brown, R.F.: A Topological Introduction to Nonlinear Analysis. Birkhäuser, Boston (2014); Addison-Wesley, Reading (1957)Google Scholar
  3. 3.
    Calabi, E.: Extremal Kähler metrics. In: Seminar on Differential Geometry. Annals of Mathematics Studies, vol. 102, pp. 259–290. Princeton University Press, Princeton (1982)CrossRefGoogle Scholar
  4. 4.
    Chen, X.X.: Weak limits of Riemannian metrics in surfaces with integral curvature bound. Calc. Var. 6, 189–226 (1998)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chen, X.X.: Extremal Hermitian metrics on Riemann surfaces. Calc. Var. Partial Differ. Equ. 8(3), 191–232 (1999)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chen, X.X.: Obstruction to the existence of metric whose curvature has umbilical Hessian in a K-surface. Commun. Anal. Geom. 8(2), 267–299 (2000)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chen, Q., Wu, Y.Y.: Existences and explicit constructions of HCMU metrics on \(S^2\) and \(T^2\). Pac. J. Math. 240(2), 267–288 (2009)CrossRefGoogle Scholar
  8. 8.
    Chen, Q., Wu, Y.Y.: Character 1-form and the existence of an HCMU metric. Math. Ann. 351(2), 327–345 (2011)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chen, Q., Chen, X.X., Wu, Y.Y.: The structure of HCMU metric in a K-surface. Int. Math. Res. Not. 2005(16), 941–958 (2005)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Chen, Q., Wu, Y.Y., Xu, B.: On one-dimensional and singular Calabi’s extremal metrics whose Gauss curvatures have nonzero umblical Hessians. Isr. J. Math 208, 385–412 (2015)CrossRefGoogle Scholar
  11. 11.
    Garcia, C.B., Li, T.Y.: On the number of solutions to polynomial systems of equations. SIAM J. Numer. Anal. 17(4), 540–546 (1980)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Hsu, S.B.: Ordinary Differential Equations with Applications, 2nd edn. World Scientific, Singapore (2013)CrossRefGoogle Scholar
  13. 13.
    Lin, C.S., Zhu, X.H.: Explicit construction of extremal Hermitian metric with finite conical singularities on \(S^2\). Commun. Anal. Geom. 10(1), 177–216 (2002)CrossRefGoogle Scholar
  14. 14.
    Springer, G.: Introduction to Riemann Surfaces. Addison-Wesley, Reading (1957)zbMATHGoogle Scholar
  15. 15.
    Wang, G.F., Zhu, X.H.: Extremal Hermitian metrics on Riemann surfaces with singularities. Duke Math. J 104, 181–210 (2000)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Wei, Z.Q., Wu, Y.Y.: Non-CSC extremal Kähler metrics on \({{S}}^2_{\{2,2,2\}}\). Results Math. 74, 58 (2019)CrossRefGoogle Scholar
  17. 17.
    Wei, Z.Q., Wu, Y.Y.: One existence theorem for Non-CSC extremal K\(\ddot{a}\)hler metrics with singularities on \(S^{2}\). TJM 22(1), 55–62 (2018)Google Scholar
  18. 18.
    Wei, Z.Q., Wu, Y.Y.: Multi-valued holomorphic functions and non-CSC extremal Kähler metrics with singularities on compact Riemann surfaces. Differ. Geom. Appl. 60(10), 66–79 (2018)CrossRefGoogle Scholar
  19. 19.
    Troyanov, M.: Prescrbing curvature on compact surface with conical singularities. Tran. Am. Math. Soc. 324(2), 793–821 (1991)CrossRefGoogle Scholar
  20. 20.
    Yau, S.T.: Calabi’s conjecture and some new results in algebraic. Proc. Natl. Acad. Sci. USA 74(5), 1798–1799 (1977)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Yau, S.T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampàre equation I. Commun. Pure Appl. Math. 31(3), 339–411 (1978)CrossRefGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2019

Authors and Affiliations

  1. 1.School of Mathematical and StatisticsHenan UniversityKaifengPeople’s Republic of China
  2. 2.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingPeople’s Republic of China

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