Green's functions and complex Monge-Ampère equations (Q6593653)

The Cheng-Li inequality in Riemannian geometry gives conditions for a uniform lower bound on Green's function in [\textit{S.-Y. Cheng} and \textit{P. Li}, Comment. Math. Helv. 56, 327--338 (1981; Zbl 0484.53034)]. In contrast to Cheng-Li's result, in the present paper the lower bounds for Green's function do not depend directly on the Ricci curvature, but rather assume integral bounds on the volume form and some specific derivatives. It seems that the authors' lower bounds for the Green's function are the same as Cheng-Li's, but it should be noted that the authors' method is completely different.\N\NIn order to show the lower bounds, a priori estimates for the general complex Monge-Ampère equation are needed. Although it is known that S.-T. Yau obtained the first estimate for solving the Calabi conjecture in [\textit{S.-T. Yau}, Commun. Pure Appl. Math. 31, 339--411 (1978; Zbl 0369.53059)], one should consider more complicated Monge-Ampère equations, which may be degenerate or singular in many different senses. From the point of view of many applications, it is important to obtain estimates that remain uniform since the Kähler class may degenerate; the authors study the cases involving both fixed and degenerating Kähler classes. In particular, the authors establish sharp \(C^1\) and \(C^2\) estimates for the complex Monge-Ampère equation, which essentially depend only on integral bounds for the right-hand side. Unfortunately, the standard maximum principle cannot be applied to the elliptic differential inequalities satisfied by the derivatives of the solution of the Monge-Ampère equation. Instead, the authors use the new lower bound for Green's function. The proof relies on an auxiliary Monge-Ampère equation and is fundamentally nonlinear. By using the Green's function estimates, the authors improve the \(C^3\) estimates of the complex Monge-Ampère equations by a weaker dependence on the right-hand side.\N\NThe main result of this paper implies that the Green's function is bounded from below if the scalar curvature is bounded from below, and the \(L^q\) norm of the volume form is bounded for some \(q>1\). The lower bound assumption for the scalar curvature is much harder to work with than the lower bound assumption for the Ricci curvature. The work of the authors in the present paper is a fundamental step towards the study of the lower bound assumption for the scalar curvature.