Geometric Non-linear problems and stability in invariant theory

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The purpose of this research is to find a necessary and sufficient condition for the existence of Kaehler metrics of constant scalar curvature in terms of stability in geometric invariant theory. There is a description of stability expressed using moment maps on symplectic manifolds. Namely orbits having a zero of the moment map are the stable orbits. From this view point it is important to study the space of all complex structures compatible with a given symplectic form, which is an infinite dimensional symplectic manifold and of which the scalar curvature is the moment map. I developed a perturbation theory of this fact and studied the infinite dimensional symplectic manifold from perturbed structures. Secondly T.Mabuchi proved that asymptotic Chow stability is equivalent to asymptotic Hilbert scheme stability.