COMPLETE MANIFOLDS WITH POSITIVE SPECTRUM

• Main Authors: Jen-Hung Cheng, 鄭任宏
• Other Authors: Ai-Nung Wang
• Format: Others
• Language: en_US
• Published: 2014
• Online Access: http://ndltd.ncl.edu.tw/handle/64193822323053413873

碩士 === 國立臺灣大學 === 數學研究所 === 102 === Let us first recall (see [12] and [10]) that an end E of a complete manifold M is non-parabolic means that E admits a positive Green''s function with Neumann boundary condition. In the previous work of the Peter Li and Tam in [12], they proved that the number of nonparabolic ends of a complete manifold is bounded from above by the dimension of the space of bounded harmonic functions with nite Dirichlet integral. On the other hand, in the work of Cao-Shen-Zhu [4] (see Corollary 4 in [13]), they observed that if an end of a manifold has a positive lower bound for the spectrum of the Laplacian, then the end must either be non-parabolic or has finite volume.These two facts together allow us to estimate the number of ends with infinite volume on a manifold with positive spectrum by estimating the dimension of the space of bounded harmonic functions with finite Dirichlet integral.Unfortunately, we do not yet know how to estimate the dimension of this space. However, we managed to estimate those harmonic functions which were constructed in the proof of the Li-Tam theorem [12]. This is sufficient to estimate the number of infinite volume ends. Let us give an outline of the proof for the theorem of Li-Tam. For our purpose, let us assume that M has at least 2 non-parabolic ends, otherwise there is nothing to prove.Suppose R_0 > 0 is sufficiently large so that M dyBp(R_0) has at least 2 disjoint non-parabolic ends E_1 and E_2. For the sake of convenience, if E is an end of M then let us denote E(R) = E∩Bp(R) and bdyE(R) = E∩bdyBp(R). We will constructa nonconstant bounded harmonic function with finite Dirichlet integral adopted for the end E_1. For R≧R_0 we will solve the following Laplace equation with the given boundary value. Let f_R be the solution of f_R = 0 on Bp(R), f_R = 1 on bdyE_1(R), and f_R = 0 on bdyBp(R) Since R≧R_0, clearly bdyE_2(R)(bdyBp(R)E_1) . Due to the assumption that both E_1 and E_2 are non-parabolic, the sequence of functions {f_R} must have a subsequence that converges to a harmonic function f defined on M which has the property that supMf = supE_1f = 1 andinfMf = inf_E_if = 0 for any non-parabolic ends E_i with i = 1. In particular, f is bounded and also it has finite Dirichlet integral. Obviously, we can use this construction on each non-parabolic end and obtain as many linearly independent harmonic functions, including the constant function, as the number of non-parabolic ends. Let us denote the space spanned by those harmonic functions by K.By estimating the dimension of K, we will be able to estimate the number of non-parabolic ends. If M, or all its ends, have positive λ_1, then dimK >number of infinite volume ends. In the following lemma, we will first obtain a decay estimate for the functions in K. Throughout the rest of the paper, we will denote the volume of the set E(R) by V_E(R) and the area of bdyE(R) by A_E(R). Recall that λ_1(E), the bottom of the L^2 spectrum of the Laplacian on E satisfying Dirichlet boundary conditions on bdyE, may be characterized alternatively as λ_1(E)∫_EΦ^2≦∫_E{|▽Φ|}^2 for all compactly supported smooth function Φ on E.