Equations of Monge-Ampère type on compact Hermitian manifolds.
Let \((X,g)\) be a compact Hermitian manifold. For \(\varphi \in C^2 (X) \), let \(M (\varphi) = \text{det} (\delta^\lambda_\mu + \nabla^\lambda_\mu \varphi)\). The author studies equations of the form \[ M (\varphi) = F(x, \nabla \varphi; \varphi), \] where \(F \in C^\infty\) \((TX \times \mathbb{R})\) is an everywhere strictly positive function with exponential growth in \(|\varphi |\) which is bounded by a power of the norm of the gradient of \(\varphi\). See also 1. \textit{S. T. Yau}, Commun. Pure Appl. Math. 31, 339-411 (1978; Zbl 0369.53059); 2. \textit{P. Delanoe}, J. Funct. Anal. 40, 358-386 (1981; Zbl 0466.58029); 3. \textit{T. Aubin}, Nonlinear analysis on manifolds. Monge-Ampère equations, Springer-Verlag (New York--Berlin 1982; Zbl 0512.53044); 4. \textit{W. Y. Ding} and \textit{G. Tian}, Invent. Math. 110, No. 2, 315-335 (1992; Zbl 0779.53044).
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