Space of K\"ahler metrics (IV)--On the lower bound of the K-energy

TL;DR

This paper investigates the relationship between constant scalar curvature K"ahler (cscK) metrics and geodesic rays, confirming a conjecture that links the existence of cscK metrics to the non-degeneracy of certain geodesic rays and establishing lower bounds for K energy functionals in specific configurations.

Contribution

It confirms a conjecture relating cscK metrics and geodesic rays, and proves lower bounds for K energy in simple test configurations with cscK central fibers.

Findings

No degenerated geodesic rays taming bounded geometry exist if cscK metrics are present.

K energy functionals have uniform lower bounds in nearby fibers of certain test configurations.

Partial confirmation of Donaldson's conjecture on K"ahler metrics.

Abstract

We partially confirm an old conjecture of Donaldson that if there exists a cscK metrics in a given K\"ahler 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 K\"ahler class.