完備流形第一固有值之上界估計

Abstract

假設M 是一個體積為無窮之完備黎曼流形, ­是在 M 上的一個緊致集. 分別在體積成長跟Ricci 曲率的下界的條件下, 去估計(M \ Omega ­) 之第一固有值的上界. 研究方法主要是根據二次微分方程解的漸近行為跟max-min principle 及Bishop 比較定理.

Let M be a complete Riemannian manifold with infnite volume and ­ be a compact subdomain in M. In this thesis we obtain two upper bound estimates for the first eigenvalue of the Laplacian on the punctured manifold M \ Omega ­ subject to volume growth and lower bound of Ricci curvature, respectively. The proof hinges on asymptotic behavior of solutions of second order differential equations, the max-min principle and Bishop volume comparison theorem.