On the existence and nonexistence of extremal metrics on toric Kähler surfaces

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• 作者：Xu-jia Wang^a ; ¹ ; ^{Xu-Jia.Wang@anu.edu.au} ; Bin Zhou^b ; ^a ; ² ; ^{bzhou@pku.edu.cn}
• 关键词：Toric surfaces ; Extremal metric ; K-stability
• 刊名：Advances in Mathematics
• 出版年：2011
• 出版时间：20 March 2011
• 年：2011
• 卷：226
• 期：5
• 页码：4429-4455
• 全文大小：222 K

文摘

In this paper we study the existence of extremal metrics on toric Kähler surfaces. We show that on every toric Kähler surface, there exists a Kähler class in which the surface admits an extremal metric of Calabi. We found a toric Kähler surface of 9 -fixed points which admits an unstable Kähler class and there is no extremal metric of Calabi in it. Moreover, we prove a characterization of the K-stability of toric surfaces by simple piecewise linear functions. As an application, we show that among all toric Kähler surfaces with 5 or 6 -fixed points, is the only one which allows vanishing Futaki invariant and admits extremal metrics of constant scalar curvature.