A gradient estimate in the Calabi–Yau theorem

By Zbigniew Blocki. 2009

84 citations recorded by the source corpus.

We prove a C1-estimate for the complex Monge–Ampère equation on a compact Kähler manifold directly from the C0-estimate, without using a C2-estimate. This was earlier done only under additional assumption of non-negative bisectional curvature.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ca96e56cefa75a9d21e2924379415f79bf98bbf8

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

On estimates for the Fu–Yau generalization of a Strominger system (2015, unknown): “} − 2 n e−ugℓm¯ {∂ ℓF∂m¯ u+∂m¯ F∂ℓu}+4αe −ug˜¯ ℓmR m¯ℓp ¯q∂ pu∂¯qu. (7.9) We now make use of a key partial cancellation, observed by Blocki in his proof of C1 estimates for the Monge-Amp`ere equation [2] (see also [9, 29], and [17, 18] for other applications of this partial cancellation), between |D”
Fu–Yau Hessian equations (2018, unknown): “ow combine this identity with the Cauchy-Schwarz inequality, which will lead to a partial cancellation of terms. This idea is also used to derive a C1 estimate for the complex Monge-Amp`ere equation, [2, 14, 24, 25, 33]. 1 |∇u|2 g |∇∇u|2 Fg ≥ gj¯i∇u ju¯i |∇u|2 g 2 F (5.21) = 1 |∇u|4 g |gj¯iu j∇u¯i| ”
Complex Monge-Ampere equations and totally real submanifolds (2009, methodology): “nstruct some special local coordinates; see Lemma 2.1. These local coordinates are crucial to our estimate of ∆uin Section 3 where we also derive gradient estimates, extending the arguments of Blocki [9] and P.-F. Guan [37] in the K¨ahler case. Section 4 concerns the boundary estimates for second de”
Geometric estimates for complex Monge–Ampère equations (2017, background): “otential functions with respect to the new metric instead of a ﬁxed reference metric for geometric complex Monge-Amp`ere equations such as those studied in Theorem 1.1. We refer interested readers to [3, 21] for gradient estimates for complex Monge-Amp`ere equations with respect to various backgroun”
The complex Monge–Ampère equation on compact Kähler manifolds (2010, background): “estimate of △φ. But we shall need this estimate. Such an estimate, which also holds for (1.3) and the Dirichlet problem of the complex Monge-Amp`ere equation on manifolds with boundary, is derived by [34, 15, 9, 32, 31, 44] in various settings; in particular, Blocki [9] established the gradient esti”
The Dirichlet problem for degenerate complex Monge-Ampere equations (2009, methodology): “bound for the bisectional curvature of the metric Ωε 0 and assume Bε > 1. Set as before Fε 1 = 2sup|∇F 1 m |. Deﬁne the function γ(x) to be γ(x) = (B +Fε 1 +1)x− 1 x+C . (3.5) Note that, unlike in [B08], this choice of γ(x) does not involve an upper bound for Φε, which is not available in the pre”