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

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.