$C^{2,\alpha }$ estimates for nonlinear elliptic equations in complex and almost complex geometry

We describe how to use the perturbation theory of Caffarelli to prove Evans-Krylov type $C^{2,\alpha }$ estimates for solutions of nonlinear elliptic equations in complex geometry, assuming a bound on the Laplacian of the solution. Our results can be used to replace the various Evans-Krylov type arguments in the complex geometry literature with a sharper and more unified approach. In addition, our methods extend to almost-complex manifolds, and we use this to obtain a new local estimate for an equation of Donaldson.