在1982年，S.T.Yau猜测：在n+1维单位球面上一个n维紧致嵌入极小超曲面的第一特征值是维数n。本项目主要是申请人通过处理Reilly公式来研究单位球面中极小超曲面的第一特征值的Yau的猜想。. 黎曼流形中超曲面的第一特征值的研究是黎曼几何中最古老和最活跃的课题之一。申请人通过考虑Dirichlet问题(1)和(2)处理Reilly公式，希望部分解决Yau的猜想。首先，申请人 在应用Dirichlet问题(1)处理Reilly公式时，引入一个新的Dirichlet问题，从而找到Dir ichlet问题(1)的解之间的一个积分关系，部分解决了Yau的猜想(见申请人报告正文中定理7)。其次，申请人应用Dirichlet问题(2)处理Reilly公式来解决了周振荣[Z]提出的问题(见申请人报告正文中定理9)。Yau的猜想目前还在研究中。

In 1982，S.T. Yau posed the conjecture：the first eigenvalue of an n-dimension compact embedded minimal hypersurface in an (n+1) dimensional unit sphere is just its dimension n。The applicant mainly deals with Reilly formula to investigate Yau’s conjecture of the first eigenvalue of minimal hypersurface in a unit sphere。 . The first eigenvalue of hypersurfaces of Riemannian manifold is one of the oldest and most active subject in Riemannian geometry。The applicant deals with the Reilly formula by considering Dirichlet problems (1) and (2) to partially solve Yau’s conjecture。Firstly，when we deal with the Reilly formula by considering the Dirichlet problem (1)， we give a new Dirichlet problem and find an integral relation of the solution of the Dirichlet problem (1)。Hence, we partially solve Yau’s conjecture(see Theorem 7 in report)。Secondly, the applicant deals with the Reilly formula by considering the Dirichlet problem (2) to solve a problem being proposed by zhou zhen rong[Z](see Theorem 9 in report)。Work of Yau’s conjecture is continuing。