Orbifold Spectral Theory
Orbifold Spectral Theory
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轨道谱理论
DOI：
10.1216/rmjm/1008959678
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发表时间：
2001
期刊：
影响因子：
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通讯作者：
Carla Farsi
中科院分区：
文献类型：
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作者：
Carla Farsi
在本文中，我们研究平滑封闭可定向黎曼轨道的索博列夫空间。特别是，我们证明了 Sobolev 嵌入定理、RellichKondrakov 定理和庞加莱不等式。从这些定理我们得出拉普拉斯谱的性质。特别是，Weil 的渐近公式和拉普拉斯特征值的下面估计与流形情况进行了类比证明。 0. 简介。在本文中，我们研究了封闭可定向轨道折叠的谱理论。 （在文献中，轨道折叠也称为 Vmanifold。）轨道折叠 Hilbert Sobolev 空间 H k 首次由 Jiang 在 [5] 中引入。 [12] 中还考虑了其他环折 Sobolev 空间。在定义了闭可定向轨道折叠的一般Sobolev空间后，我们建立了Sobolev嵌入定理和RellichKondrakov定理。通过使用这些定理，与流形情况类似，我们证明了轨道拉普拉斯特征值的韦尔渐近公式。我们还证明了庞加莱不等式。我们对该材料的介绍紧密遵循[9]、[1]，它们处理多种情况。通过证明更精确的索博列夫不等式，我们还可以从拉普拉斯算子特征值的下方获得估计，将[4]的结果推广到轨道情况。本文是一个正在进行的项目的起点，该项目旨在将流形谱理论的几个众所周知的结果推广到轨道折叠。现在我们回顾一下本文 [10]、[5]、[7] 中使用的一些基本定义。除非另有说明，否则我们所有的轨道折叠都被假定为光滑的和黎曼的。封闭的可定向轨道折叠 M 可以由有限数量的图表 (Ωl, φl)l=1,... ,N 覆盖，其中 Ωl = Ωl/Gl，其中 Ωl 同胚于 R 和 Gl，是 SO(n) 的有限子群。假设图表变化的局部提升是平滑的。编辑于 1999 年 8 月 27 日收到。版权所有 ©2001 Rocky Mountain Mathematics Consortium 215
In this paper we study Sobolev spaces for smooth closed orientable Riemannian orbifolds. In particular we prove the Sobolev embedding theorem, the RellichKondrakov theorem and Poincare’s inequalities. From these theorems we derive properties of the spectrum of the Laplacian. In particular, Weil’s asymptotic formula and estimates from below of the eigenvalues of the Laplacian are proved in analogy with the manifold case. 0. Introduction. In this paper we study spectral theory for closed orientable orbifolds. (In the literature orbifolds are also called Vmanifolds.) Orbifold Hilbert Sobolev spaces H k were first introduced by Chiang in [5]. Other orbifold Sobolev spaces are also considered in [12]. After defining general Sobolev spaces for closed orientable orbifolds, we establish Sobolev embedding theorems and the RellichKondrakov theorem. By using these theorems we prove, in analogy with the manifold case, Weil’s asymptotic formula for the eigenvalues of the orbifold Laplacian. We also prove Poincare’s inequalities. Our presentation of this material follows closely [9], [1], which deal with the manifold case. By proving more refined Sobolev inequalities we also obtain estimates from below of the eigenvalues of the Laplacian, generalizing the results of [4] to the orbifold case. This paper is the starting point of an ongoing project aiming at generalizing several well-known results of spectral theory for manifolds to orbifolds. We will now recall a few basic definitions used throughout the paper [10], [5], [7]. Unless otherwise specified, all our orbifolds are assumed to be both smooth and Riemannian. A closed orientable orbifold, M , can be covered by a finite number of charts (Ωl, φl)l=1,... ,N , where Ωl = Ωl/Gl with Ωl homeomorphic to R and Gl a finite subgroup of SO(n). The local lifts of the changes of charts are assumed to be smooth. Received by the editors on August 27, 1999. Copyright c ©2001 Rocky Mountain Mathematics Consortium 215
Calabi 型泛函的最优下界
DOI：
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发表时间：
2020
期刊：
影响因子：
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作者：
Kubo Tomoya;Matsuoka Eiich;Kotegawa Hisashi;Tou Hideki;Nakamura Ai;Aoki Dai;Harima;T. Hisamoto
通讯作者：
T. Hisamoto