Variational problems for Riemannian functionals and arithmetic groups
Variational problems for Riemannian functionals and arithmetic groups
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黎曼泛函和算术群的变分问题
DOI：
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发表时间：
1997
期刊：
影响因子：
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通讯作者：
S. Weinberger
中科院分区：
文献类型：
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作者：
A. Nabutovsky;S. Weinberger
本文提出了一种新的方法来解决维数为n >= 5的固定紧流形M^n上黎曼结构（即黎曼度量的等距类）空间Riem（M^n）上的变分问题。这种方式常常使一个取代认为变分问题Riem (M ^ n)(或者在某些子集Riem (M ^ n))同样的问题,但在空间Riem (n ^ n)为每一个歧管n ^ n类的紧凑型阀组相同的尺寸和相同的同源性M ^ n但以下两个有用的属性:(1)如果任何黎曼结构\ν歧管n ^ n从这个类,这样Ric_ (n ^ n \ν)> = - (n - 1),然后的体积(n ^ n, \ν)大于1;(2)该类流形不承认非负标量曲率的黎曼度量。
In this paper we introduce a new approach to variational problems on the space Riem(M^n) of Riemannian structures (i.e. isometry classes of Riemannan metrics) on any fixed compact manifold M^n of dimension n >= 5. This approach often enables one to replace the considered variational problem on Riem(M^n) (or on some subset of Riem(M^n)) by the same problem but on spaces Riem(N^n) for every manifold N^n from a class of compact manifolds of the same dimension and with the same homology as M^n but with the following two useful properties: (1) If \nu is any Riemannian structure on any manifold N^n from this class such that Ric_(N^n,\nu) >= -(n-1), then the volume of (N^n,\nu) is greater than one; and (2) Manifolds from this class do not admit Riemannian metrics of non-negative scalar curvature.
As a first application we prove a theorem which can be informally explained as follows: Let M be any compact connected smooth manifold of dimension greater than four, M et(M) be the space of isometry classes of compact metric spaces homeomorphic to M endowed with the Gromov-Hausdorff topology, Riem_1(M) in M et(M ) be the space of Riemannian structures on M such that the absolute values of sectional curvature do not exceed one, and R_1(M) denote the closure of Riem_1(M) in M et(M ). Then diameter regarded as a functional on R_1(M) has infinitely many "very deep" local minima.