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Recently Darvas–Rubinstein proved a convergence result for the Kähler–Ricci iteration, which is a sequence of recursively defined complex Monge–Ampére equations. We introduce the Monge–Ampére iteration to be an analogous, but in a sense more general, sequence of recursively defined real Monge–Ampére second boundary value problems, and we establish sufficient conditions for its convergence. Each step in the iteration is a carefully chosen optimal transportation problem. We determine two cases where the convergence conditions are satisfied and provide geometric applications for both. First, we give a new proof of Darvas and Rubinstein’s general theorem on the convergence of the Ricci iteration in the case of toric Kähler manifolds, while at the same time generalizing their theorem to general convex bodies. Second, we introduce the affine iteration to be a sequence of prescribed affine normal problems and prove its convergence to an affine sphere. These give a new approach to recent existence and uniqueness results due to Berman–Berndtsson and Klartag.
KeywordsReal Monge–Ampére equation Kähler geometry Kähler–Einstein manifolds Ricci iteration Affine differential geometry Optimal transport
Mathematics Subject ClassificationPrimary 32Q20 Secondary 53A15
Many thanks go to Y.A. Rubinstein for suggesting the Monge–Ampére iteration as a fruitful topic of study and for his indispensable guidance, inspiration, and encouragement; and to T. Darvas for his insights into the Kähler–Ricci iteration and for welcoming all my questions along the way. This research was also supported by the BSF Grant 2012236 and the NSF Grants DMS-1515703 and DMS-1440140.
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