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


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 \(S^{2}\).

This is a preview of subscription content, access via your institution.


  1. 1.

    Aubin, T.: Nonlinear Analysis on Manifolds, Monge Ampre Equations. Grundlehren der Mathematicschen Wissenchaften, vol. 252. Springer, New York (1982)

    Google Scholar 

  2. 2.

    Brown, R.F.: A Topological Introduction to Nonlinear Analysis. Birkhäuser, Boston (2014); Addison-Wesley, Reading (1957)

  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)

  4. 4.

    Chen, X.X.: Weak limits of Riemannian metrics in surfaces with integral curvature bound. Calc. Var. 6, 189–226 (1998)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Chen, X.X.: Extremal Hermitian metrics on Riemann surfaces. Calc. Var. Partial Differ. Equ. 8(3), 191–232 (1999)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    Article  Google 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)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Hsu, S.B.: Ordinary Differential Equations with Applications, 2nd edn. World Scientific, Singapore (2013)

    Google 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)

    Article  Google Scholar 

  14. 14.

    Springer, G.: Introduction to Riemann Surfaces. Addison-Wesley, Reading (1957)

    Google Scholar 

  15. 15.

    Wang, G.F., Zhu, X.H.: Extremal Hermitian metrics on Riemann surfaces with singularities. Duke Math. J 104, 181–210 (2000)

    MathSciNet  Article  Google 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)

    Article  Google 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)

    Article  Google Scholar 

  19. 19.

    Troyanov, M.: Prescrbing curvature on compact surface with conical singularities. Tran. Am. Math. Soc. 324(2), 793–821 (1991)

    Article  Google 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)

    MathSciNet  Article  Google 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)

    Article  Google Scholar 

Download references


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).

Author information



Corresponding author

Correspondence to Zhiqiang Wei.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Wei, Z., Wu, Y. On the Existence of Non-CSC Extremal Kähler Metrics with Finite Singularities on \(S^2\). J Geom Anal 31, 1555–1567 (2021). https://doi.org/10.1007/s12220-019-00315-y

Download citation


  • Extremal Kähler metric
  • Conical singularities
  • Cusp singularities

Mathematics Subject Classification

  • 53C21
  • 53C56