Monge-Amp\`ere Iteration

TL;DR

This paper introduces the Monge-Ampere iteration, a generalization of complex Monge-Ampere equations, establishing convergence conditions and applying them to prove new results in Kähler geometry and affine differential geometry.

Contribution

It generalizes the Ricci iteration convergence results to real Monge-Ampere problems and introduces the affine iteration for affine spheres, providing new proofs and broader applicability.

Findings

Convergence conditions for the Monge-Ampere iteration are established.

A new proof of Ricci iteration convergence on toric Kähler manifolds is provided.

The affine iteration converges to an affine sphere, offering a new proof of Klartag's existence result.

Abstract

In a recent paper, Darvas-Rubinstein proved a convergence result for the Kahler-Ricci iteration, which is a sequence of recursively defined complex Monge-Ampere equations. We introduce the Monge-Ampere iteration to be an analogous, but more general, sequence of recursively defined real Monge-Ampere second boundary value problems, and we establish sufficient conditions for its convergence. We then determine two cases where these conditions are satisfied and provide geometric applications for both. First, we give a new proof of Darvas and Rubinstein's theorem on the convergence of the Ricci iteration in the case of toric Kahler 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, giving a new approach to an existence result due to Klartag.