  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.
