# On sharp lower bounds for Calabi type functionals and destabilizing properties of gradient flows

## TL;DR

This paper establishes sharp lower bounds for Calabi type functionals on K"ahler manifolds using a metric approach, extending Donaldson's conjecture and constructing explicit minimizers, with applications to Fano manifolds.

## Contribution

It proves a metric analogue of Donaldson's conjecture by enlarging the test configuration space and replacing invariants with the radial Mabuchi K-energy, demonstrating the bound's sharpness and explicit minimizers.

## Key findings

- Proved a sharp lower bound for Calabi energy using geodesic rays and radial Mabuchi K-energy.
- Constructed explicit minimizers of the Mabuchi K-energy functional.
- Extended the results to Ricci-Calabi energy on Fano manifolds.

## Abstract

Let $X$ be a compact K\"ahler manifold with a given ample line bundle $L$. In \cite{Don05}, Donaldson proved that the Calabi energy of a K\"ahler metric in $c_1(L)$ is bounded from below by the supremum of a normalized version of the minus Donaldson--Futaki invariants of test configurations of $(X,L)$. He also conjectured that the bound is sharp. In this paper, we prove a metric analogue of Donaldson's conjecture, we show that if we enlarge the space of test configurations to the space of geodesic rays in $\mathcal{E}^2$ and replace the Donaldson--Futaki invariant by the radial Mabuchi K-energy $\mathbf{M}$, then a similar bound holds and the bound is indeed sharp. Moreover, we construct explicitly a minimizer of $\mathbf{M}$. On a Fano manifold, a similar sharp bound for the Ricci--Calabi energy is also derived.