A gradient estimate for the Monge-Ampère equation on compact almost Hermitian manifolds
Let \((X,\omega_0)\) be a closed Kähler manifold. One can formulate a complex Monge-Ampère equation \( (\omega_0 + \frac{i}{2}\partial\overline{\partial}\varphi)^n = e^{f(z)}\omega_0^n\), where \(f(z)\) is a given smooth function satisfying some necessary conditions. The existence of solutions to this equation, famously proven by Yau via the continuity method, establishes the Calabi conjecture and asserts the existence of a unique Ricci-flat Kähler metric when \(X\) has vanishing first Chern class. The work of \textit{J. Chu} et al. [J. Eur. Math. Soc. (JEMS) 21, No. 7, 1949--1984 (2019; Zbl 1452.32035)] establishes a priori estimates when \(X\) is a compact, almost complex manifold. Further work by \textit{B. Guo} et al. [``On \(L^\infty\) estimates for complex Monge-Ampère equations'', Preprint, \url{arXiv:2106.02224}] establishes sharp \(L^\infty\) estimates without pluripotential theory, and uses the Alexandrov-Bakelmann-Pucci (ABP) maximum principle to recover gradient estimates established in [\textit{D. H. Phong} et al., Surv. Differ. Geom. 17, 327--410 (2012; Zbl 1382.32023)]. The paper under review generalizes the above work by establishing gradient estimates in the setting when \((X,J,g)\) is almost Hermitian. That is, when \(g(X,Y) = g(JX,JY)\) for any pair of vector fields \(X, Y\). The paper proves this by again utilizing the ABP maximum principle and explicit computation in local coordinates.
scientific article; zbMATH DE number 7467755
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