On a uniform estimate for the quaternionic Calabi problem

Quaternionic Monge-Ampere equation and Calabi problem for HKT-manifolds

A quaternionic version of the Calabi problem on Monge-Ampere equation is introduced. It is a quaternionic Monge-Ampere equation on a compact hypercomplex manifold with an HKT-metric. The equation is non-linear elliptic of second order. For a hypercomplex manifold with holonomy in SL(n;H), uniqueness (up to a constant) of a solution is proven, as well as the zero order a priori estimate. The existence of solution is conjectured, similar to Calabi-Yau theorem. We reformulate this quaternionic equation as a special case of a complex Hessian equation, making sense on any complex manifold.

Solvability of the quaternionic Monge-Ampere equation on compact manifolds with a flat hyperKaehler metric

A quaternionic version of the Calabi problem was recently formulated by M. Verbitsky and the author. It conjectures a solvability of a quaternionic Monge-Ampere equation on a compact HKT manifold (HKT stays for HyperKaehler with Torsion). In this paper this problem is solved under an extra assumption that the manifold admits a flat hyperKaehler metric compactible with the underlying hypercomplex structure. The proof uses the continuity method and a priori estimates.

超埃尔米特流形上四元数蒙日-安培方程的精确一致界

Sharp uniform bound for the quaternionic Monge-Ampere equation on hyperhermitian manifolds

本文利用Guo和Phong的最新方法，为超埃尔米特流形上的四元数蒙日-安培方程及一系列相关四元数偏微分方程提供了精确的 $C^0$ 一致估计，改进了先前对右端范数和初始度量条件的依赖。

一致 Skoda 可积性与 Calabi-Yau 退化

Uniform Skoda integrability and Calabi-Yau degeneration

本文研究 Calabi-Yau 流形的极化代数退化，通过证明一致 Skoda 型估计和一致 $L^\infty$-估计作出贡献。

Higher-order estimates for collapsing Calabi-Yau metrics

We prove a uniform C^alpha estimate for collapsing Calabi-Yau metrics on the total space of a proper holomorphic submersion over the unit ball in C^m. The usual methods of Calabi, Evans-Krylov, and Caffarelli do not apply to this setting because the background geometry degenerates. We instead rely on blowup arguments and on linear and nonlinear Liouville theorems on cylinders. In particular, as an intermediate step, we use such arguments to prove sharp new Schauder estimates for the Laplacian on cylinders. If the fibers of the submersion are pairwise biholomorphic, our method yields a uniform …

坍缩 Calabi-Yau 度量的光滑渐近性

Smooth asymptotics for collapsing Calabi-Yau metrics

本文研究紧 Calabi-Yau 流形沿全纯纤维化坍缩时的度量行为，建立了远离奇异纤维的全阶先验估计与按纤维直径展开的渐近公式。