A lower bound for λ1 on manifolds with boundaryon manifolds with boundary
A lower bound for λ1 on manifolds with boundaryon manifolds with boundary
复制标题
具有边界的流形上 λ1 的下界
DOI：
10.1007/bf02564440
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发表时间：
1987
期刊：
影响因子：
--
通讯作者：
A. Derdzinski
中科院分区：
文献类型：
--
作者：
C. Croke;A. Derdzinski
本文的目的是在紧黎曼流形（M，g）上建立hi的一个几何下界，其中~（-1）=~ q（M，g）是在8 M（Dirichlet边界条件）上为零的函数的（正）Laplacian的第一特征值.对于单位切球面Ux中x ~ 9 Int M的v，设l（v）<-~是Int M中与v相切的最大测地线yo的长度，用dvx表示Ux上的标准测度，用cr（n-1）表示单位（n-1）球面的体积，我们证明了：
The aim of this paper is to establish a geometric lower bound for hi on compact Riemannian manifolds (M, g) with boundary,~-1=~ q (M, g) being the first eigenvalue of the (positive) Laplacian on functions that vanish on 8M (Dirichlet boundary condition). We also classify the manifolds for which equality in our estimate is attained.For v in the unit tangent sphere Ux at x 9 Int M, let l (v)<-~ be the length of the maximal geodesic yo in Int M, tangent to v. Denoting by dvx the canonical measure on Ux and by cr (n-1) the volume of the unit (n-1)-sphere, we prove