C^1,1 regularity for degenerate complex Monge–Ampère equations and geodesic rays

Abstract

In this paper, we prove a C 1,1 estimate for solutions of complex Monge-Ampère equations on compact almost Hermitian manifolds. Using this C 1,1 estimate, we show existence of C 1,1 solutions to the degenerate Monge-Ampère equations, the corresponding Dirichlet problems and the singular Monge-Ampère equations. We also study the singularities of the pluricomplex Green's function. In addition, the proof of the above C 1,1 estimate is valid for a kind of complex Monge-Ampère type equations. As a geometric application, we prove the C 1,1 regularity of geodesics in the space of Sasakian metrics.

On the "Equation missing" Regularity of Geodesics in the Space of Kähler Metrics

“…By [7,Theorem 6.1], v 0 has minimal singularities, so there exists a large constant C such that ψ − C v 0 v ε for all ε > 0. The proofs of Lemma 2.2 and Proposition 2.5 now apply directly (One can also probably use easier proofs to obtain (3.3), but it follows immediately from what we have done already -see also [12,Section 4]).…”

“…Proof of Theorem 1.1. Our strategy will be the same as in [28], which is a combination of the techniques in [2] and the estimates in [12]. We begin by approximating V by the following envelopes:…”

“…We may then apply the maximum principle argument in [12, Lemma 4.3] using ψ instead of ψ, which gives us the estimate everywhere, as desired. Note that, although it is assumed in [12] that ψ has analytic singularities, it is easy to see that the proof only needs the weaker assumption of exponential smoothness, in the specific form of (2.2).…”

“…In [27], Mabuchi introduced a Riemannian structure on the space of Kähler metrics on a compact manifold X without boundary. Later, Semmes [34] and Donaldson [20] independently showed that these geodesics could be given as solutions to the Dirichlet problem for the complex Monge-Ampère operator, and since then there has been a great deal of work to establish regularity and positivity properties of such solutions -see [3,4,11,12,13,15,24,31,32,33,35]. In particular, the recent work of Chu-Tosatti-Weinkove [11] establishes C 1,1 regularity of solutions (based on their earlier work [10]), which is known to be optimal by examples of Lempert-Vivas [30], Darvas-Lempert [18], and Darvas [14].…”

Envelopes with Prescribed Singularities

“…Examples in [3,Example 5.2], [16,Example 7.2], and [41,Corollary 4.7] show that one cannot in general expect P (u) ∈ C 2 , and so the best possible regularity is C 1,1 , which was indeed shown by Tosatti [42] and Chu-Zhou [20] when [θ] is Kähler. More recently, C 1,1 regularity of P (u) on compact subsets of the ample locus of [α] has been proved by Chu-Tosatti-Weinkove [19] when [α] is nef and big. These results are obtained by combining the "zero temperature limit" deformation of Berman [4] with the C 1,1 estimates for degenerate complex Monge-Ampère equations developed by Chu-Tosatti-Weinkove in [17,18].…”