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.

arXiv comment40 pages. v2. extends convergence result to general Fano manifolds and includes other minor changes

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Cited by in corpus (12)

• Moment maps, nonlinear PDE, and stability in mirror symmetry
• Mabuchi Solitons and Relative Ding Stability of Toric Fano Varieties
• Relative Ding Stability and an Obstruction to the Existence of Mabuchi Solitons
• Geometric Partial Differential Equations from Unified String Theories
• New curvature flows in complex geometry
• Collapsing of the line bundle mean curvature flow on Kähler surfaces
• Geometric flow, Multiplier ideal sheaves and Optimal destabilizer for a Fano manifold
• 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