On the existence and nonexistence of extremal metrics on tonic Kahler surfaces

In this paper we study the existence of extremal metrics on tonic Kahler surfaces. We show that on every toric Kahler surface, there exists a Kahler class in which the surface admits an extremal metric of Calabi. We found a tonic Kahler surface of 9 T-C(2)-fixed points which admits an unstable Kahler class and there is no extremal metric of Calabi in it. Moreover, we prove a characterization of the K-stability of tonic surfaces by simple piecewise linear functions. As an application, we show that among all tonic Kahler surfaces with 5 or 6 T-C(2)-fixed points, CP2#3 (CP) over bar (2) is the only one which allows vanishing Futaki invariant and admits extremal metrics of constant scalar curvature. (C) 2010 Elsevier Inc. All rights reserved.