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

Abstract

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.

26 pages, v2 final version, to appear in Calc. Var. PDE

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• Multiplier Ideal Sheaves and the Kähler-Ricci Flow
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• On estimates for the Fu-Yau generalization of a Strominger system
• The Dirichlet Problem of Fully Nonlinear Equations on Hermitian Manifolds
• Transverse Fully Nonlinear Equations on Sasakian Manifolds and Applications
• A parabolic Monge-Ampère type equation of Gauduchon metrics
• Generalized complex Monge-Ampère type equations on closed Hermitian manifolds
• Second order estimates for complex Hessian equations on Hermitian manifolds
• Concavity of the Lagrangian Phase Operator and Applications
• Deformed Hermitian-Yang-Mills Equation on Compact Hermitian Manifolds
• A flow of conformally balanced metrics with Kähler fixed points
• The deformed Hermitian-Yang-Mills equation on almost Hermitian manifolds
• The parabolic flows for complex quotient equations
• Leafwise flat forms on Inoue-Bombieri surfaces
• Parabolic complex Monge-Ampere equations on compact Kahler manifolds
• A special Lagrangian type equation for holomorphic line bundles
• Monge-Ampère type equations on almost Hermitian manifolds
• Positivity in Foliated Manifolds and Geometric Applications
• The Aleksandrov-Bakelman-Pucci estimate and the Calabi-Yau equation
• The Dirichlet problem for a complex Hessian equation on compact Hermitian manifolds with boundary
• Local regularity for concave homogeneous complex degenerate elliptic equations comparable to the Monge-Ampère equation
• The complex Monge-Ampère equation on some compact Hermitian manifolds
• The continuity equation, Hermitian metrics and elliptic bundles
• Fully non-linear elliptic equations on compact almost Hermitian manifolds