Monge-Ampère operators, energy functionals, and uniqueness of Sasaki-extremal metrics

Abstract

We develop some pluripotential theoretic techniques for the transversally holomorphic foliation of a Sasakian manifold. We prove the convexity of the K-energy along weak geodesics for Sasakian manifolds. This implies that the K-energy is bounded below if a constant scalar curvature structure exists with those metrics minimizing it. More generally, a relative version of the K-energy is convex, and bounded below if there exists a Sasaki-extremal metric, providing an important necessary condition for Sasaki-extremal metrics. Another application is a proof of the uniqueness of Sasaki-extremal metrics, for a fixed transversally holomorphic structure on the Reeb foliation.

40 pages, some minor editing of the first version

References in corpus (6)

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• On deformation of extremal metrics
• On the existence of constant scalar curvature Kähler metric: a new perspective
• Sasakian Geometry, Hypersurface Singularities, and Einstein Metrics
• Convexity of the K-energy on the space of Kahler metrics and uniqueness of extremal metrics
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Cited by in corpus (6)

• Sasaki-Einstein metrics and K-stability
• Weighted K-stability of polarized varieties and extremality of Sasaki manifolds
• Transverse Fully Nonlinear Equations on Sasakian Manifolds and Applications
• Scalar curvature and properness on Sasaki manifolds
• Geometrical pluripotential theory on Sasaki manifolds
• Greatest lower bounds on the transverse Ricci curvature of some toric Sasaki manifolds