Stability estimates for the complex Monge-Ampère and Hessian equations

Abstract

A new proof for stability estimates for the complex Monge-Ampère and Hessian equations is given, which does not require pluripotential theory. A major advantage is that the resulting stability estimates are then uniform under general degenerations of the background metric in the case of the Monge-Ampère equation, and under degenerations to a big class in the case of Hessian equations.

On the modulus of continuity of solutions to complex Monge-Ampère equations

“…We remark that the estimate (1.2) continues to hold when the function e F is not smooth. This can be seen by a smoothing argument combined with the stability estimate of complex Monge-Ampère equations [12,8] and Theorem 1. The continuity of u when e F is not smooth had been obtained by Kolodziej [11] using pluripotential theory and an argument by contradiction.…”

“…As in [7,8,9], we aim to compare ψ s,k with the solution u to (1.1). Consider the following test function…”

“…Next we apply a De Giorgitype iteration argument (c.f. [11,7,8]). Choose a δ 0 > 0 small which depends only on n, p, ω 0 , e F L 1 ( log L) p such that for all δ ∈ (0, δ 0 ], it follows from Lemma 3…”

On $L^\infty$ estimates for Monge-Ampère and Hessian equations on nef classes

“…The goal of this short note is to show that the PDE approach introduced in [12,13] for L ∞ and Trudinger-type estimates for general classes fully non-linear equations on a compact Kähler manifold applies as well to Monge-Ampère and Hessian equations on nef classes.…”

“…The key to the approach in [12,13] is an estimate of Trudinger-type, obtained by comparing the solution ϕ of the given equation to the solution of an auxiliary Monge-Ampère equation with the energy of the sublevel set function −ϕ + s on the right hand side. We shall see that, in the present case of nef classes, the argument can still be made to work by replacing ϕ by ϕ − V , where V is the envelope of the nef class.…”