We introduce the inverse Monge-Ampere flow as the gradient flow of the Ding energy functional on the space of Kahler metrics in $2 \pi \lambda c_1(X)$ for $\lambda=\pm 1$. We prove the long-time existence of the flow. In the canonically polarized case, we show that the flow converges smoothly to the unique Kahler-Einstein metric with negative Ricci curvature. In the Fano case, assuming $X$ admits a Kahler-Einstein metric, we prove the weak convergence of the flow to a Kahler-Einstein metric. In general, we expect that the limit of the flow is related with the optimally destabilizing test configuration for the $L^2$-normalized non-Archimedean Ding functional. We confirm this expectation in the case of toric Fano manifolds.

中文翻译：

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Monge-Ampère 逆流及其对 Kähler-Einstein 度量的应用

我们将反蒙格-安培流作为在 $2 \pi \lambda c_1(X)$ for $\lambda=\pm 1$ 的 Kahler 度量空间上的 Ding 能量泛函的梯度流引入。我们证明了流的长期存在。在规范极化的情况下，我们表明流动平滑地收敛到具有负 Ricci 曲率的唯一 Kahler-Einstein 度量。在 Fano 的情况下，假设 $X$ 承认 Kahler-Einstein 度量，我们证明流向 Kahler-Einstein 度量的弱收敛。一般来说，我们期望流动的限制与$L^2$-归一化的非阿基米德丁泛函的最优不稳定测试配置有关。我们在复曲面法诺流形的情况下证实了这种期望。