CR流形上拉普拉斯算子的第一正特征值研究

流形上Laplace算子的第一正特征值与流形的几何性质密切相关.本项目主要研究CR流形上Sub-Laplace算子与Kohn-Laplace算子的第一正特征值的估计.CR流形上的Lichnerowicz-Obata结论大部分已得到论证,而3维情形下关于Kohn-Laplace算子的Obata结论仍有待解决;Pseduo-Hermitian流形上Laplace算子谱的下确界的上限估计至今鲜有结果.基于此,本项目拟解决以下两个问题:首先,刻画一个一般3维Pseudo-Hermitian流形,使其Paneitz算子非负,进而得到一般3维Pseudo-Hermitian流形上关于Kohn-Laplace算子的Obata结论;其次,探究Pseudo-Hermitian流形上Sub-Laplace算子的第一正特征值的精确上界及刚性定理.上述工作将有助于进一步认识与研究CR流形.

The first positive eigenvalues for Laplacians on manifolds are closely related to the geometric properties of such manifolds. The purpose of this project is to study the estimations on first positive eigenvalues for Sub-Laplacians and Kohn-Laplacians on CR manifolds.Most of the lichnerowicz-obata type conclusions on CR manifolds have been proved but the Obata type conclusion on a general Pseudo-Hermitian manifold with dimension 3. On the other hand, there are few results on estimation for the upper bound of the first positive eigenvalue for Laplacians on the Pseudo-Hermitian manifolds. Therefore, this project aims to solve the following problems:Firstly, characterize a general Pseudo-Hermitian manifold such that its Paneitz operator is always non-negative. Thus one can obtian the Obata type conclusion for Kohn-Laplacians on a general Pseudo-Hermitian manifold with dimension 3.Secondly, study the upper bound of the first positive eigenvalue for Sub-Laplacians on the Pseudo-Hermitian manifolds and the corresponding rigidity theorem.The above work will contribute to further understanding and study of CR manifolds.