The $C^0$ estimate for the quaternionic Calabi conjecture

A uniform estimate for general quaternionic Calabi problem (with appendix by Daniel Barlet)

We prove a $C^0$ a priori estimate on a solution of the quaternionic Calabi problem on an arbitrary compact connected HKT-manifold. This generalizes earlier works where this result was proven under certain extra assumptions on the manifold.

On a uniform estimate for the quaternionic Calabi problem

We establish a C^0 a priori bound on the solutions of the quaternionic Calabi-Yau equation (of Monge-Ampere type) on compact HKT manifolds with a locally flat hypercomplex structure. As an intermediate step, we prove a quaternionic version of the Gauduchon theorem.

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.

$L^\infty$ 估计：具有指定体积形式的四元数 Gauduchon 度量的势函数

$L^\infty$ estimate for the potential of quaternionic Gauduchon metric with prescribed volume form

本文受四元数 Calabi 猜想启发，在紧致 $SL(n,\mathbb{H})$ 流形上引入四元数 Gauduchon 猜想，研究具有指定体积形式的四元数 Gauduchon 度量的存在性，将其转化为二阶完全非线性椭圆方程并建立势函数的 $L^\infty$ 估计。

The Calabi-Yau conjectures for embedded surfaces

In this paper we will prove the Calabi-Yau conjectures for embedded surfaces. In fact, we will prove considerably more. The Calabi-Yau conjectures about surfaces date back to the 1960s. Much work has been done on them over the past four decades. In particular, examples of Jorge-Xavier from 1980 and Nadirashvili from 1996 showed that the immersed versions were false; we will show here that for embedded surfaces, i.e., injective immersions, they are in fact true.

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.