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

Abstract

Let $X$ be a compact Kähler manifold with a given ample line bundle $L$. Donaldson proved an inequality between the Calabi energy of a Kähler metric in $c_1\left(L\right)$ and the negative of normalized Donaldson–Futaki invariants of test configurations of $\left(X,L\right)$. He also conjectured that the bound is sharp.

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 $\mathit{M}$, then a similar bound holds and the bound is indeed sharp. Moreover, we construct explicitly a minimizer of $\mathit{M}$. On a Fano manifold, a similar sharp bound for the Ricci–Calabi energy is also derived.