Ricci-flat metrics on the complexification of a compact rank one symmetric space
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We construct a complete Ricci-flat Kähler metric on the complexification of a compact rank one symmetric space. Our method is to look for a Kähler potential of the form ψ = ƒ(u), whereu satisfies the homogeneous Monge-Ampère equation. We use the high degree of symmetry present to reduce the non-linear partial differential equation governing the Ricci curvature to a simple second-order ordinary differential equation for the functionf. To prove that the resulting metric is complete requires some techniques from symplectic geometry.
KeywordsSymmetric Space Ricci Curvature Cotangent Bundle Stein Manifold Real Projective Space
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