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

Abstract

We introduce the inverse Monge-Ampere flow as the gradient flow of the Ding energy functional on the space of Kahler metrics in $2\pi \lambda c_1(X)$ 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.

40 pages. v2. extends convergence result to general Fano manifolds and includes other minor changes

References in corpus (3)

• Mabuchi metrics and properness of the modified DING functional
• Orthogonal projection of a test configuration to vector fields
• Generalized Kähler Einstein metrics and uniform stability for toric Fano manifolds

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• Tan-concavity property for Lagrangian phase operators and applications to the tangent Lagrangian phase flow
• A flow of conformally balanced metrics with Kähler fixed points
• Parabolic complex Monge-Ampere equations on compact Kahler manifolds
• Parabolic complex Monge-Ampère equations on compact Hermitian manifolds
• Hessian of the Ricci Calabi functional