We consider fully nonlinear elliptic equations on complex manifolds which depend on the gradient in some nontrivial ways. Some of these equations arise from interesting problems in complex geometry, such as a conjecture by Gauduchon which is a natural generalization of Calabi conjecture to the Hermitian setting, and finding balanced metrics on Hermitian manifolds. We shall discuss difficulties in solving such equations and present recent results in our attempt to overcome these difficulties.  Our goal is to establish some general existence results which we hope will find useful applications in complex geometry in the near future. We'll explain how our results provide a proof to the Gauduchon conjecture building on previous work of Tossati-Weinkove and others. The talk is based on joint work with Xiaolan Nie, Chunhui Qiu and Rirong Ruan. 

We survey some new and old, positive and negative results on a priori estimates, regularity, and rigidity for special Lagrangian equations with or without certain convexity. The "gradient" graphs of solutions are minimal or maximal Lagrangian submanifolds, respectively in Euclidean or pseudo-Euclidean spaces. In the latter pseudo-Euclidean setting, these equations are just Monge-Ampere equations. Development on the parabolic side (Lagrangian mean curvature flows) will also be mentioned.

We complete the proof of the main theorem by showing that if X^3 is isomorphic to X, then X^{\omega} has a parity-reversing automorphism. By our previous results this implies X^2 is isomorphic to X as well. The proof generalizes to show that for any n > 1, if X^n is isomorphic to X, then X^2 is isomorphic to X. Time permitting we will discuss related results, including the existence of an A and X such that A^2X is isomorphic to X, while AX is not.