Stability of solutions to complex Monge–Ampère flows

Abstract

We establish a stability result for elliptic and parabolic complex Monge-Ampère equations on compact Kähler manifolds, which applies in particular to the Kähler-Ricci flow.Dedicated to Jean-Pierre Demailly on the occasion of his 60th birthday.

Stability and Hölder regularity of solutions to complex Monge–Ampère equations on compact Hermitian manifolds

“…The proof of Theorem 1.1 goes along the same lines as in [38]. An immediate consequence of Theorem 1.1 is a stability estimate for the constant c: Corollary 1.2.…”

Continuous solutions to Monge–Ampère equations on Hermitian manifolds for measures dominated by capacity

“…Very recently Lu et al [30] improved the stability estimates for f , g ∈ L p (X ) with p > 1, with the same exponent as in the Kähler case, by a clever use of the stability estimate for Monge-Ampère type equation from the Guedj et al work [17]. Given the existence of solution in Corollary 3.3 for measures belonging to F (X , h) with L 1 -densities we expect that the stability estimate above can be improved.…”

Pluripotential Kähler–Ricci flows

“…Step 2. We finally remove the smoothness assumption on the data and the Lipschitz condition on ϕ by using the stability result above together with an argument from [GLZ18].…”

“…Proof. We use a perturbation argument as in [GLZ18] which goes back to the work of Ko lodziej [Ko l96]. For convenience we normalize θ so that…”

“…It is delicate to compare pluripotential and viscosity concepts in general. We refer the interested reader to [GLZ3] where we prove, when g is continuous, that the viscosity solution constructed in [EGZ16] coincides with the pluripotential solution Φ(g, F, ω t , ϕ 0 ).…”