#### Vol. 14, No. 6, 2021

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

### Mingchen Xia

Vol. 14 (2021), No. 6, 1951–1976
##### 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 ${\mathsc{ℰ}}^{2}$ and replace the Donaldson–Futaki invariant by the radial Mabuchi K-energy $M$, then a similar bound holds and the bound is indeed sharp. Moreover, we construct explicitly a minimizer of $M$. On a Fano manifold, a similar sharp bound for the Ricci–Calabi energy is also derived.

##### Keywords
destabilizing property, weak Calabi flow, inverse Monge–Ampère flow, geodesic ray
##### Mathematical Subject Classification 2010
Primary: 32Q15, 32U05, 51F99
Secondary: 35K96