Space of Kähler metrics III--On the lower bound of the Calabi energy and geodesic distance

Abstract

In this paper, we first prove a folklore conjecture on a greatest lower bound of the Calabi energy in all Kähler manifold. Similar result in algebriac setting was obtained by S. K. Donaldson. Secondly, we give an upper/lower bound estimate of the K energy in terms of the geodesic distance and the Calabi energy. This is used to prove a theorem on convergence of Kähler metrics in holomorphic coordinates, with uniform bound on the Ricci curvature and the diameter. Thirdly, we set up a framework for the existence of geodesic rays when an asymptotic direction is given. I

References in corpus (3)

• Interior estimates for solutions of Abreu's equation
• The Monge-Ampère operator and geodesics in the space of Kähler potentials
• An energy-theoretic approach to the Hitchin-Kobayashi correspondence for manifolds, II

Cited by in corpus (9)

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• The space of volume forms
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• On Pointwise Gradient Estimates for the Complex Monge-Ampere Equation
• The K-energy on small deformations of constant scalar curvature Kahler manifolds
• Note on geodesic rays tamed by simple test configurations
• Remarks on the existence of bilaterally symmetric extremal Kähler metrics on ${\mathbb{CP}}^2♯2\overline{{\mathbb{CP}}^2}$