Smoothness of the curvature of an HCMU on S2 or T2

doi:10.7523/j.issn.2095-6134.2008.5.002

›› 2008, Vol. 25 ›› Issue (5): 585-591.DOI: 10.7523/j.issn.2095-6134.2008.5.002

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Smoothness of the curvature of an HCMU on S² or T²

Wu Ying-yi

School of Mathematical Sciences, Graduate University of the Chinese Academy of Sciences, Beijing 100049, China

Received:1900-01-01 Revised:1900-01-01 Online:2008-09-15

Abstract

Abstract:

An HCMU is a kind of extremal metric with singularities on a Riemann surface. If the area and Calabi energy are both bounded, the Gauss curvature of an HCMU is a continuous function on the Riemann surface. In this paper we get an explicit construction of an HCMU on S² which has no saddle point of the Gauss curvature of the metric. Further more we prove that on S² or T² the Gauss curvature of an HCMU is smooth if and only all of the singular angles are integers.

Key words: extremal metric, HCMU, conical singularity, singular angle

Wu Ying-yi. Smoothness of the curvature of an HCMU on S² or T²[J]. , 2008, 25(5): 585-591.

[1]	GUO Jinyu, WU Yingyi, WEI Zhiqiang. Conformal metrics on Riemann surfaces with cusp singularities [J]. , 2016, 33(6): 721-728.
[2]	WEI Zhiqiang, WU Yingyi, GUO Jinyu. Existence of a class HCMU metric on S² [J]. , 2016, 33(5): 577-583.
[3]	WEI Zhiqiang, WU Yingyi. An existence theorem and energy integral formula of HCMU metrics [J]. , 2016, 33(1): 16-22.
[4]	WU Ying-Yi. On the character 1-form of an HCMU metric [J]. , 2009, 26(2): 185-193.

Smoothness of the curvature of an HCMU on S² or T²

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