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We introduce the inverse Monge-Ampere flow as the gradient flow of the Dingenergy 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 thecanonically polarized case, we show that the flow converges smoothly to theunique Kahler-Einstein metric with negative Ricci curvature. In the Fano case,assuming $X$ admits a Kahler-Einstein metric, we prove the weak convergence ofthe flow to a Kahler-Einstein metric. In general, we expect that the limit ofthe flow is related with the optimally destabilizing test configuration for the$L^2$-normalized non-Archimedean Ding functional. We confirm this expectationin the case of toric Fano manifolds.

The inverse Monge-Ampere flow and applications to Kahler-Einstein metrics