C_2,α regularities and estimates for nonlinear elliptic and parabolic equations in geometry

We give sharp C_2,α estimates for solutions of some fully nonlinear elliptic and parabolic equations in complex geometry and almost complex geometry, assuming a bound on the Laplacian of the solution. We also prove the analogous results to complex Monge–Ampère equations with conical singularities. As an application, we obtain a local estimate for Calabi–Yau equation in almost complex geometry. We also improve the C_2,α regularities and estimates for viscosity solutions to some uniformly elliptic and parabolic equations. All our results are optimal regarding the Hölder exponent.