$$C^{2,\alpha }$$C2,α estimates for nonlinear elliptic equations in complex and almost complex geometry

By Valentino Tosatti, Yu Wang, B. Weinkove, Xiaokui Yang. 2014

75 citations recorded by the source corpus.

We describe how to use the perturbation theory of Caffarelli to prove Evans–Krylov type $$C^{2,\alpha }$$C2,α 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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/9c881ad81ffdf53b9d0daf0ab59f020e8adaf9dd

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Concavity of the Lagrangian phase operator and applications (2016, methodology): “g [37] in the study of the complex Monge-Amp`ere equation. The extension argument is used to extend the concave operator Gfrom the Hermitian matrices to the symmetric matrices. We refer the reader to [37, 31] and, for instance, the proof of [6, Lemma 6.1], but leave the details to the interested rea”
Degenerate complex Hessian equations on compact K\"ahler manifolds (2014, background): “partial diﬀerential equations the complex Hessian equation is an interesting and important object. Recently another general (and powerful) C2,αestimate for equations of this type has been obtained in [41]. In [1] equations of complex Hessian type appear in the study of quaternionic geometry. Thus it”
The Monge–Ampère equation for non-integrable almost complex structures (2016, methodology): “section, we complete the proofs of the main results stated in the introduction. Proof of Theorem 1.2. Thanks to Propositions 3.1 and 5.1, we can obtain a priori C2,α estimates from the main result of [44], for some 0 <α<1 (and in fact for all 0 <α<1 by [10]). Higher order estimates follo”
Fully non-linear elliptic equations on compact Hermitian manifolds (2015, methodology): “ this case we have limt→∞ f(λ+ tei) = ∞ for all i. In particular, the same argument as the proof of Proposition 21 implies the following, which answers a question raised in Tosatti-Wang-Weinkove-Yang [36]. FULLY NON-LINEAR ELLIPTIC EQUATIONS ON COMPACT HERMITIAN MANIFOLDS 31 Proposition 24. Let (M,α”
Fu–Yau Hessian equations (2018, methodology): “e is therefore bounded in Cβ(X). Since ˆσ1/2 2 (χ) is a concave uniformly elliptic operator on the space of admissible solutions, we may apply a Evans-Krylov type result of Tosatti-Weinkove-Wang-Yang [30] to conclude kukC2,β ≤ C. However, for general k≥ 2 case, it is impossible to re-write equation ”