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

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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.

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Authors and Affiliations

  1. Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL, 60208, USA

    Valentino Tosatti, Yu Wang, Ben Weinkove & Xiaokui Yang

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  1. Valentino Tosatti
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  2. Yu Wang
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  3. Ben Weinkove
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  4. Xiaokui Yang
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Correspondence to Ben Weinkove.

Additional information

Communicated by O. Savin.

Research supported in part by NSF grants DMS-1236969, DMS-1308988 and DMS-1332196. The first named-author is supported in part by a Sloan Research Fellowship.

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Tosatti, V., Wang, Y., Weinkove, B. et al. \(C^{2,\alpha }\) estimates for nonlinear elliptic equations in complex and almost complex geometry. Calc. Var. 54, 431–453 (2015). https://doi.org/10.1007/s00526-014-0791-0

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