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

We introduce the inverse Monge-Ampère flow as the gradient flow of the Ding energy functional on the space of Kähler metrics in 2πλc₁(X) for λ = ±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 L²-normalized non-Archimedean Ding functional. We confirm this expectation in the case of toric Fano manifolds.