On the modulus of continuity of solutions to complex Monge-Ampère equations

Abstract

In this paper, we prove a uniform and sharp estimate for the modulus of continuity of solutions to complex Monge-Ampère equations, using the PDE-based approach developed by the first three authors in their approach to supremum estimates for fully non-linear equations in Kähler geometry. As an application, we derive a uniform diameter bound for Kähler metrics satisfying certain Monge-Ampère equations.

Green's functions and complex Monge-Ampère equations

“…We conclude this section by comparing the lower bound of the Green's function obtained in Theorem 2 with the classical one in Cheng-Li [3]. Let ω = ω X +i∂ ∂ϕ ∈ [ω X ] be a Kähler metric with e Fω L 1+ǫ (X,ω n X ) ≤ N. Suppose Ric(ω) ≥ −κ ′ ω for some κ ′ ≥ 0, then from [13,6] we know diam(X, ω) ≤ C(n, ω X , N). Then Cheng-Li's estimate (1.2) implies the Green's function associated with ω is bounded below.…”

“…Another approach to C 0 estimates, using PDE methods, was introduced very recently in [9]. This method can also apply to nef classes [12], and lead to many sharp estimates, including stability estimates [11], diameter estimates [13], and non-collapse estimates [14]. We have just seen it applied to lower bounds for Green's functions in the first part of this paper.…”

Local noncollapsing for complex Monge-Ampère equations

“…Finally we mention that when the function e F on the right-side of (4.1) belongs to L q for some q > 1, the diameter bound of ω ϕ has been proved in [17], using the Hölder continuity of the solution ϕ established in [16,2] (see [11]). When e F L 1 (log L) p ≤ A for some A > 0 and p > 3n, the diameter bound of ω ϕ has been obtained in [11] after they establish the modulus of continuity of the Kähler potentials. 4.2.…”

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.…”