Weak solutions to the complex Monge-Ampère equation on Hermitian manifolds (Q2786820)

Consider the complex Monge-Ampère equation \((\omega+\sqrt{-1}\partial \bar{\partial} \phi)^n=f\omega^n\) on a compact hermitian manifold. Thanks to the works of Cherrier, Tosatti-Weinkove, if \(f\) is smooth there exist unique smooth solutions to the equation. On hermitian manifolds, there is the so-called Chern-Ricci flow which behaves like the Kähler-Ricci flow. Its study requires an analysis of the Monge-Ampère equation even when \(f\) is merely in \(L^p\) for \(p>1\). This is the problem that the current paper solves. The main theorem (Theorem 0.1) asserts existence and uniqueness of continuous solutions solving the equation in the Bedford-Taylor sense. The proof of it uses a modified comparison principle (Theorem 0.2) which is a substitute for the usual comparison principle used on Kähler manifolds. The modified comparison principle is then also used to deduce \(L^p\) stability and existence results on appropriate domains in \(\mathbb{C}^n\) (Corollaries 0.3 and 0.4).NEWLINENEWLINEFor the entire collection see [Zbl 1322.53004].