The inverse Monge–Ampère flow and applications to Kähler–Einstein metrics

Abstract

We introduce the inverse Monge–Ampère flow as the gradient flow of the Ding energy functional on the space of Kähler 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 Kähler–Einstein metric with negative Ricci curvature. In the Fano case, assuming the $X$ admits a Kähler–Einstein metric, we prove the weak convergence of the flow to the Kähler–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.