We investigate the relationship between stability and the existence of extremal Kähler metrics on certain toric surfaces. In particular, we consider how log stability depends on weights for toric surfaces whose moment polytope is a quadrilateral. For quadrilaterals, we give a computable criterion for stability with 0 weights along two of the edges of the quadrilateral. This in turn implies the existence of a definite log-stable region for quadrilaterals. This uses constructions due to Apostolov-Calderbank-Gauduchon and Legendre.

This work was done as a part of the author's PhD thesis at Imperial College London. I would like to thank my supervisor Simon Donaldson for his encouragement and insight. I gratefully acknowledge the support from the Simons Center for Geometry and Physics, Stony Brook University at which some of the research for this paper was performed. I would also like to thank Vestislav Apostolov for helpful comments. Finally, I thank the referee for careful reading of the manuscript and many useful suggestions for improvement.