$I$-properness of Mabuchi's $K$-energy

TL;DR

This paper establishes conditions for the properness of Mabuchi's K-energy by analyzing energy functionals and geodesic rays in the space of Kähler metrics, using gradient flow convergence.

Contribution

It provides new criteria for the properness of Mabuchi's K-energy based on energy bounds and geodesic ray analysis in Kähler geometry.

Findings

Lower bound of the energy functional tilde E^eta established

Criteria for geodesic rays to detect bounds of tilde  J^eta functional developed

Convergence of the gradient flow of tilde  J^eta functional demonstrated

Abstract

Over the space of K\"ahler metrics associated to a fixed K\"ahler class, we first prove the lower bound of the energy functional $\tilde E^\beta$, then we provide the criterions of the geodesics rays to detect the lower bound of $\tilde {\mathfrak J}^\beta$-functional. They are used to obtain the properness of Mabuchi's $K$-energy. The criterions are examined by showing the convergence of the negative gradient flow of $\tilde {\mathfrak J}^\beta$-functional.