New applications of the Ahlfors Laplacian

爱因斯坦流形上Chen-Nagano规范的光谱性质

Spectral Properties of the Chen-Nagano Gauge on Einstein Manifolds

本文引入基于Chen-Nagano规范的新谱几何框架，建立迹与无迹部分的一阶和二阶谱耦合关系，证明在负曲率爱因斯坦流形上该规范退化为经典横无迹规范，并给出Ricci流下的动力学解释。

An Obata type result for the first eigenvalue of the sub-Laplacian on a CR manifold with a divergence free torsion

We prove a CR Obata type result that if the first positive eigenvalue of the sub-Laplacian on a compact strictly pseudoconvex pseudohermitian manifold with a divergence free pseudohermitian torsion takes the smallest possible value then, up to a homothety of the pseudohermitian structure, the manifold is the standart Sasakian unit sphere. We also give a version of this theorem using the existence of a function with traceless horizontal Hessian on a complete, with respect to Webster's metric, pseudohermitian manifold.

极值Ricci-pinched $G_2$-结构的性质与扰动

Properties and Perturbations of Extremally Ricci-Pinched $G_2$-Structures

本文研究紧致7维流形上极值Ricci-pinched $G_2$-结构的性质，给出其等价刻画，证明自同构群有限，并构造保持该条件的扰动。

广义相对论真空约束方程研究中超曲面第二基本形式的$L^2$-分解

$L^2$-decomposition of the second fundamental form of a hypersurface in the study of the general relativistic vacuum constraint equations

本文利用Ahlfors Laplacian研究洛伦兹时空中闭合类空超曲面第二基本形式的$L^2$-正交分解，并应用于分析广义相对论真空约束方程的代数-微分性质。

The Obata sphere theorems on a quaternionic contact manifold of dimension bigger than seven

We prove a quaternionic contact versions of the Obata's sphere theorems. We show that if the first positive eigenvalue of the sub-Laplacian on a compact qc manifold of dimension bigger than seven takes the smallest possible value then, up to a homothety of the qc structure, the manifold is qc equivalent to the standard 3-Sasakian sphere. We also give a version of the theorem on non-compact qc manifold which is complete with respect to the associated Riemannian metric using the existence of a function with traceless horizontal Hessian. A qc version of the Liouville theorem is shown for qc-confo…