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February, 2018 One Existence Theorem for non-CSC Extremal Kähler Metrics with Conical Singularities on $S^2$
Zhiqiang Wei, Yingyi Wu
Taiwanese J. Math. 22(1): 55-62 (February, 2018). DOI: 10.11650/tjm/8086

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 following question: if we give $N$ points $p_1, \ldots, p_N$ on $S^2$ and $N$ positive real numbers $2\pi \alpha_1, \ldots, 2\pi \alpha_N$ with $\alpha_n \neq 1$, $n = 1, \ldots, N$, what condition can guarantee the existence of a non-CSC HCMU metric which has conical singularities $p_1, \ldots, p_N$ with singular angles $2\pi \alpha_1, \ldots, 2\pi \alpha_N$ respectively. We prove that if there are at least $N-2$ integers in $\alpha_1, \ldots, \alpha_N$ then there exists one non-CSC HCMU metric on $S^2$ satisfying the condition stated above no matter where the given points are.

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Zhiqiang Wei. Yingyi Wu. "One Existence Theorem for non-CSC Extremal Kähler Metrics with Conical Singularities on $S^2$." Taiwanese J. Math. 22 (1) 55 - 62, February, 2018. https://doi.org/10.11650/tjm/8086

Information

Received: 20 October 2016; Revised: 14 April 2017; Accepted: 18 April 2017; Published: February, 2018
First available in Project Euclid: 17 August 2017

zbMATH: 06965359
MathSciNet: MR3749354
Digital Object Identifier: 10.11650/tjm/8086

Subjects:
Primary: 53C21, 53C55, 58D17

Rights: Copyright © 2018 The Mathematical Society of the Republic of China

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Vol.22 • No. 1 • February, 2018
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