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

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-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 $C^{2,\alpha }$ regularities and estimates for viscosity solutions to some uniformly elliptic and parabolic equations. All our results are optimal regarding the Hölder exponent.

22 pages, title changed, added parabolic version of main theorem (see section 5). This article has been accepted for publication on Calculus of Variations and Partial Differential Equations. This is accepted version. The final publication is available at Springer via http://dx.doi.org/10.1007/s00526-015-0948-5

References in corpus (2)

• Taming symplectic forms and the Calabi-Yau equation
• A priori estimates for Donaldson's equation over compact Hermitian manifolds

Cited by in corpus (3)

• Transverse Fully Nonlinear Equations on Sasakian Manifolds and Applications
• A parabolic Monge-Ampère type equation of Gauduchon metrics
• Leafwise flat forms on Inoue-Bombieri surfaces