The Monge–Ampère equation for non-integrable almost complex structures

Abstract

We show existence and uniqueness of solutions to the Monge-Ampère equation on compact almost complex manifolds with non-integrable almost complex structure.

Transverse fully nonlinear equations on Sasakian manifolds and applications

“…By the argument in the proof [61, Proposition 6] (based on Proposition 4.2), we can find a uniform constant κ > 0 such that at any point p ∈ M , if |λ(A †,u )| > R and A †,u = diag{λ 1 , · · · , λ n } with λ 1 ≥ · · · ≥ λ n , then there holds either Since tr ω † h † = n i=1 λ i > 0, it is sufficient to get the upper bound of λ 1 to complete the proof. Our background is a counterpart to the Kähler case in [61], and we use the method revised from [62] (see also [14,16,41,61]), where another auxiliary function ϕ(t) = D 1 e −D 2 t is introduced. Since there are no terms about the first order derivatives of u in (h † ) ij , and the adapted Chern connection is torsion free, we need not estimate as many terms as done in [62] (see also [61]).…”

Envelopes with Prescribed Singularities

“…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]. The envelope (1.2) with u = 0 is also known as the "pluricomplex Green's function," as introduced in [31] (see also [32]) with ψ(z) = log |z| 2 and in [36] for general ψ with analytic singularities, and whose regularity has been studied extensively by many authors [1,8,29,35], primarily in the case when ψ has poles at a discreet set of points.…”

C1,1 regularity for degenerate complex Monge–Ampère equations and geodesic rays

“…Recently, Chu-Tosatti-Weinkove [14] solved (1.1) on compact almost Hermitian manifolds. Unlike Kähler and Hermitian cases, almost Hermitian case is much more complicated.…”

“…Combining the maximum principle and a series of delicate calculations, the real Hessian estimate was obtained. Following the approach of [14], Chu-Tosatti-Weinkove [15] established the existence of C 1,1 solutions to the homogeneous complex Monge-Ampère equation and solved the open problem of C 1,1 regularity of geodesics in the space of Kähler metrics (see [9]). Further applications of these ideas can be found in [42,17,16,13].…”

“…However, the C 1,1 estimate in [14] depends on sup M f , sup M |∂(log f )| g and lower bound of ∇ 2 (log f ). Hence, except the homogeneous complex Monge-Ampère equation, it is impossible to apply this C 1,1 estimate in the study of degenerate complex Monge-Ampère equation.…”