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

TL;DR

This paper establishes optimal $C^{2, heta}$ regularity estimates for solutions to fully nonlinear elliptic and parabolic equations in complex and almost complex geometry, including equations with singularities, with broad applications.

Contribution

It provides sharp regularity estimates for complex geometric PDEs, extending to equations with conical singularities and improving regularity results for viscosity solutions.

Findings

Sharp $C^{2, heta}$ estimates for nonlinear elliptic and parabolic equations.

Extension of regularity results to equations with conical singularities.

Improved regularity and estimates for viscosity solutions.

Abstract

We give sharp $C^{2,\alpha}$ 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-Amp\`{e}re equations with conical singularities. As an application, we obtain a local estimate for Calabi-Yau equation in almost complex geometry. We also improve the $C^{2,\alpha}$ regularities and estimates for viscosity solutions to some uniformly elliptic and parabolic equations. All our results are optimal regarding the H\"{o}lder exponent.