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.
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Acknowledgments
The author would like to thank his advisor Gang Tian for constant encouragement and several useful comments on an earlier version of this paper. The author would also like to thank Yuanqi Wang for some helpful discussions about complex Monge–Ampère equations with conical singularities. The author would also like to thank Valentino Tosatti and Jingang Xiong for many helpful conversations. The author would also like to thank CSC for supporting the author visiting Northwestern University.
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Communicated by A. Chang.
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Chu, J. \(C^{2,\alpha }\) regularities and estimates for nonlinear elliptic and parabolic equations in geometry. Calc. Var. 55, 8 (2016). https://doi.org/10.1007/s00526-015-0948-5
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DOI: https://doi.org/10.1007/s00526-015-0948-5