On uniform estimate in Calabi-Yau theorem

Abstract

We show that the uniform estimate in the Calabi-Yau theorem easily follows from the local stability of the complex Monge-Ampère equation.

On a uniform estimate for the quaternionic Calabi problem

“…The method of the proof is heavily based on the paper [6] of Z. Blocki. We use the generic notation const to denote any positive constant that depends only on n, M, I, J, K, Ω.…”

The Calabi–Yau theorem and Kähler currents

“…In this case the Calabi-Yau theorem is replaced by its Hermitian counterpart, proved by Weinkove and the author [41] (see also [2,7,9,17,24,26,27,34,44,42] for earlier results and later developments, [19,20,21,22,30,36,45,46] for other Monge-Ampère type equations on non-Kähler manifolds, and [40] for a very recent Calabi-Yau theorem for Gauduchon metrics on Hermitian manifolds). The key new difficulty is that now in general we have to modify the function F in (1.1) by adding a constant to it, namely we obtain…”

The $\mathcal C^{2,\alpha}$ estimate of complex Monge-Ampere equation

“…Kolodziej ([29,30]) proved the existence and Hölder estimate of solution to the complex Monge-Ampère equation when the right hand side is a nonnegative L p function for p > 1. There are further existence and regularity results on the complex Monge-Ampère equation with right hand side less regular or degenerate, see references [3,13,26,4,52,43,27,20,19,22] for details.…”