| Summary: | In this thesis we study the relationship between the existence of extremal Kähler metrics and stability. We introduce a space of symplectic potentials for toric manifolds, which we show gives metrics with mixed Poincaré type and cone angle singularities. We show uniqueness and that existence implies stability for extremal metrics arising from these potentials. For quadrilaterals, we give a computable criterion for stability in certain cases, giving a definite log-stable region for generic quadrilaterals. We use computational tools to find new examples of stable and unstable toric manifolds. For Poincaré type manifolds with an S1-action, we prove a version of the LeBrun-Simanca openness theorem and Arezzo-Pacard blow-up theorem.
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