An investigation of stability on certain toric surfaces

We investigate the relationship between stability and the existence of extremal K\"ahler metrics on certain toric surfaces. In particular, we consider how log stability depends on weights for toric surfaces whose moment polytope is a quadrilateral. We introduce a space of symplectic potentials for toric manifolds, which induces metrics with mixed Poincar\'e type and cone angle singularities. 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 generic quadrilaterals. This uses constructions due to Apostolov-Calderbank-Gauduchon and Legendre.