Green's functions and complex Monge-Ampère equations

Abstract

Uniform L 1 and lower bounds are obtained for the Green's function on compact Kähler manifolds. Unlike in the classic theorem of Cheng-Li for Riemannian manifolds, the lower bounds do not depend directly on the Ricci curvature, but only on integral bounds for the volume form and certain of its derivatives. In particular, a uniform lower bound for the Green's function on Kähler manifolds is obtained which depends only on a lower bound for the scalar curvature and on an L q norm for the volume form for some q > 1. The proof relies on auxiliary Monge-Ampère equations, and is fundamentally non-linear. The lower bounds for the Green's function imply in turn C 1 and C 2 estimates for complex Monge-Ampère equations with a sharper dependence on the function on the right hand side.

Hermite--Einstein metrics in singular settings

“…• We establish the existence of Hermite-Einstein metrics for ω X -stable reflexive coherent sheaves F on X. Our method relies on a recent, very important work of Guo, Phong and Sturm [GPS22]. It is explained in Sections 1-5.…”

Uniform entropy and energy bounds for fully non-linear equations

“…Moreover, under the assumption (2.10), we will show sup X F is bounded above by a uniform constant. To see this, we begin with a mean-value type inequality which was proved in [15] for complex Monge-Ampère equations and the arguments there can be easily adapted to the current situation. But for convenience of the readers, we include a proof of this inequality in Lemma 3 below.…”

“…Proof of Lemma 3. As in [15], we may assume that N := c n ω Vω X |u|e nF ω n X ≤ 1, otherwise, we can consider the rescaled function ũ = u/N, which still satisfies the differential inequality (4.4) with the same a. We also assume {u > 0} = ∅, otherwise this lemma is trivial.…”

“…The proof of the L ∞ bound for fully non-linear PDE's which we just described was based on a comparison with an auxiliary Monge-Ampère equation involving integrals A s on the sublevel sets Ω s = {z ∈ X; ϕ < −s} of the function ϕ. This method turns out to be particularly effective: it has been extended by various authors to stability estimates [12], nef classes [13], moduli of continuity [14,18], lower bounds for the Green's function [15,17], as well as Hermitian manifolds [16] and parabolic equations [5].…”

On $L^\infty$ estimates for fully nonlinear partial differential equations

“…The strategy of the proof of Theorem 1 is to employ a local auxiliary complex Monge-Ampère equations as in [20] and a localized argument similar to that in [1]. It is now known that auxiliary Monge-Ampère equations on compact closed manifolds can be particularly effective, with recent major successes in the constant scalar curvature metric problem [5], the L ∞ estimates in the Kähler case [20,21,22,23], the corresponding treatment of parabolic equations [6], and applications to lower bounds for Green's functions [24]. However, the use of a local auxiliary Monge-Ampère equation makes the method readily applicable to many other settings.…”