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Convergence of Riemannian manifolds and spectrum of Laplacian

黎曼流形的收敛性和拉普拉斯谱

基本信息

批准号：
14540056
负责人：
SHIOYA Takashi
金额：
$ 2.18万
依托单位：
Tohoku University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2004
项目状态：
已结题

项目摘要

Let M_i→M and Y_i→Y, i=1,2,3,..., be two Gromov-Hausdorff convergent sequences of proper metric spaces, where ‘proper' means that any closed bounded set is compact. We give Radon measures on all M_i and M, and assume that the measure on M_i weakly converges to that on M. We are interested in the asymptotic behavior and convergence of maps u_i : M_i→Y_i. We introduce a concept of L^p-convergence of such u_i to a map u : M→Y,p【greater than or equal】1, and establish a theory of convergence of energy functionals E_i defined on the mapping space {u : M_i→Y_i} by generalizing Mosco's variational convergences. Mosco defined the asymptotically compactness of {E_i}, as a generalization of the Rellich compactness. The asymptotic compactness is useful to obtain the convergence of energy minimizers, i.e., harmonic maps. Under a uniform bound of the Poincare constant for E_i and some condition on the metric structure of M, we prove the asymptotic compactness of {E_i}. We say that E_i compactly converges to a functional E on {u : M→Y} if E_i Γ-converges to E and if {E_i} is asymptotically compact. We prove that the compact convergence E_i→E is equivalent to the Gromov-Hausdorff convergence of the E_i-sublevel sets to the E-sublevel sets. This gives a geometric interpretation of the compact convergence. Assume in addition that Y_i are all CAT(0)-spaces and E_i are convex and lower semi-continuous. Then, we prove that the compact convergence E_i→E is equivalent to the convergence of the corresponding resolvents, where the resolvents for E_i and E are defined by using the minimizers of the Moreau-Yosida approximation. As applications, we investigate the spectra of the Korevaar-Schoen approximating energy forms. We also obtain the compactness of the energy functionals over Riemannian manifolds under a bound of Ricci curvature.

设M_i→M， Y_i→Y, i=1,2,3，…，是两个正则度量空间的Gromov-Hausdorff收敛序列，其中“正则”表示任何闭有界集合都是紧的。我们给出了所有M_i和M上的Radon测度，并假设M_i上的Radon测度弱收敛于M上的Radon测度。我们对映射u_i: M_i→Y_i的渐近性和收敛性感兴趣。我们引入了映射u: M→Y，p【大于等于】1的L^p收敛的概念，并通过推广Mosco变分收敛，建立了定义在映射空间{u: M_i→Y_i}上的能量泛函E_i的收敛理论。Mosco定义了{E_i}的渐近紧性，作为Rellich紧性的推广。渐近紧性有助于得到能量极小值的收敛性，即调和映射。在E_i的庞加莱常数的一致界和M的度量结构的某些条件下，证明了{E_i}的渐近紧性。我们说E_i紧收敛于{u: M→Y}上的一个泛函E，如果E_i Γ-converges趋近于E，如果{E_i}是渐近紧的。证明了E_i→E的紧收敛等价于E_i-子水平集到E-子水平集的Gromov-Hausdorff收敛。这给出了紧收敛性的几何解释。另外假设Y_i都是CAT(0)-空间，E_i是凸且下半连续的。然后，我们证明了E_i→E的紧收敛性等价于对应解的收敛性，其中E_i和E的解是用Moreau-Yosida近似的极小值来定义的。作为应用，我们研究了korevar - schoen近似能量形式的谱。我们还得到了黎曼流形上能量泛函在里奇曲率界下的紧性。