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arXiv · 1712.01685

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

Tristan C. Collins·Tomoyuki Hisamoto·Ryosuke Takahashi
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Abstract

We introduce the inverse Monge-Ampere flow as the gradient flow of the Ding energy functional on the space of Kahler metrics in $2 πλc_1(X)$ for $λ=\pm 1$. 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.

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Tristan C. Collins, Tomoyuki Hisamoto, Ryosuke Takahashi. 2018-02-05. The inverse Monge-Ampere flow and applications to Kahler-Einstein metrics. https://arxiv.org/abs/1712.01685

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