A generalization of the Monge-Ampère equation on compact Hermitian manifolds
Let \((X,g)\) be a compact Hermitian manifold. If \(\varphi\in C^2(X)\), consider the changes of the Hermitian metric defined by \(g_1(\varphi)= -\varphi g+i\partial \overline{\partial}\varphi\) and \(g_2^\pm(\varphi)= g+i\partial \overline{\partial}\varphi\pm \nabla\varphi\otimes \nabla\varphi\), where each \(g_i(\varphi)\) is positive definite. In such cases, \(\varphi\) is said to be admissible. The author proves the existence of an admissible \(C^\infty\) solution to equations of the form \[ [\text{det } g_i(\varphi)] [\text{det}(g)]^{-1}= F(x,\nabla\varphi;\varphi), \] where \(F\in C^\infty(TX\times \mathbb{R})\) is an everywhere strictly positive function satisfying some growth assumptions. For some equations, it is shown that the above solution is also unique. This is an extension to the Hermitian case of results of \textit{P. Delanoë} [J. Funct. Anal. 45, 403-430 (1982; Zbl 0497.58026)].
scientific article; zbMATH DE number 889863