C1,1 regularity of degenerate complex Monge–Ampère equations and some applications

Fully non-linear degenerate elliptic equations in complex geometry

“…In this case however, we show that the corresponding non-positive terms are also quite small, picking up an extra factor of λ −2 1 compared to the other summands in (1.4). Thus, we show that these terms can be controlled by the addition of an extra quantity to the maximum principle depending on ∇ 2 u, similar to [13]. As a remark, we point out that the extra term in [13] depends on ∂∂u, and that the argument there also depends heavily on the structure of F MA .…”

“…Thus, we show that these terms can be controlled by the addition of an extra quantity to the maximum principle depending on ∇ 2 u, similar to [13]. As a remark, we point out that the extra term in [13] depends on ∂∂u, and that the argument there also depends heavily on the structure of F MA . In contrast, our new extra term depends on ∇ 2 u, making the argument significantly more delicate.…”

“…Previous arguments for the Monge-Ampère equation ( [17,13]) heavily utilize special properties of the corresponding F . More precisely, for the Monge-Ampère equation, both F ii and F ij,kl have simple expressions in terms of the solution metric -in the general context of Equation (1.1), this is no longer true however, so we need to apply new techniques to control (1.4).…”