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 $M$, 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.

让 $X$ 是具有给定充足线丛的紧凑 Kähler 流形 $升$. 唐纳森证明了 Kähler 度量的卡拉比能量之间的不等式$C_1\left(升\right)$ 以及测试配置的归一化 Donaldson-Futaki 不变量的否定 $\left(X,升\right)$. 他还推测界限是尖锐的。

我们证明了唐纳森猜想的度量模拟；我们表明，如果我们将测试配置的空间扩大到测地线的空间${\mathcal{ℰ}}^2$ 并用径向 Mabuchi K 能量代替 Donaldson-Futaki 不变量 $米$，那么类似的界限成立并且界限确实很尖锐。此外，我们明确地构造了一个最小化器$米$. 在 Fano 流形上，也导出了 Ricci-Calabi 能量的类似锐界。