Abstract
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 TC2-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 TC2-fixed points, CP2#3CP¯2 is the only one which allows vanishing Futaki invariant and admits extremal metrics of constant scalar curvature.
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