Space of Kähler metrics (IV)--On the lower bound of the K-energy

Abstract

We partially confirm an old conjecture of Donaldson that if there exists a cscK metrics in a given Kähler class, then there is no degenerated geodesic ray which is tamed by a bounded ambient geometry unless it parallels to a holomorphic line consists of cscK metrics only. We also prove that for simple test configuration where the central fibre has a cscK metric, the K energy functionals in the nearby fibre must also have a uniform lower bound in its underlying Kähler class.

References in corpus (6)

• 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
• K-stability of constant scalar curvature Kähler manifolds
• Test configurations and Geodesic rays
• Note on geodesic rays tamed by simple test configurations

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• $\cF$-functional and geodesic stability
• Calabi flow, Geodesic rays, and uniqueness of constant scalar curvature Kähler metrics
• Constant scalar curvature equation and the regularity of its weak solution
• Convergence of Kähler-Ricci flow on Fano manifolds, II
• Pseudo-Calabi Flow
• On the structure of almost Einstein manifolds
• Kähler non-collapsing, eigenvalues and the Calabi flow
• The K-energy on small deformations of constant scalar curvature Kahler manifolds
• A splitting theorem for extremal Kaehler metrics
• Energy functionals and Kähler-Ricci solitons
• Extremal Kähler metrics and energy functionals on projective bundles
• On the extension and smoothing of the Calabi flow on complex tori