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We study the Calabi functional on a ruled surface over a genus two curve. For polarizations which do not admit an extremal metric we describe the behavior of a minimizing sequence splitting the manifold into pieces. We also show that the Calabi flow starting from a metric with suitable symmetry gives such a minimizing sequence.
We show that if a polarised manifold admits an extremal metric then it is K-polystable
relative to a maximal torus of automorphisms.
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