The Steklov spectrum and coarse discretizations of manifolds with boundary
The Steklov spectrum and coarse discretizations of manifolds with boundary
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DOI：
10.4310/pamq.2018.v14.n2.a3
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发表时间：
2016-12
期刊：
影响因子：
--
通讯作者：
B. Colbois;A. Girouard;B. Raveendran
中科院分区：
文献类型：
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作者：
B. Colbois;A. Girouard;B. Raveendran
考虑一类紧致n维黎曼流形，其边界为圆柱形，Ricci曲率由给定的常数下界，内射半径由正的常数下界，远离边界.对于这个类的流形M，我们引入一个概念的离散化，导致一个图形的边界大致等距M，常数只取决于尺寸和边界上的曲率和内射半径。在这方面，我们证明了一个统一的谱比较不等式的Steklov特征值的流形M和它的离散化。给出了它在构造具有定长边界和任意大Steklov谱隙的曲面序列中的一些应用。特别是，我们得到这样的序列的表面与连接的边界。这些应用是基于从具有良好扩展特性的图序列中获得的图状表面的构造。
We consider the class of compact n-dimensional Riemannian manifolds with cylindrical boundary, Ricci curvature bounded below by a given constant and injectivity radius bounded below by a positive constant, away from the boundary. For a manifold M of this class, we introduce a notion of discretization, leading to a graph with boundary which is roughly isometric to M, with constants depending only on the dimension and bounds on curvature and injectivity radius. In this context, we prove a uniform spectral comparison inequality between the Steklov eigenvalues of the manifold M and those of its discretization. Some applications to the construction of sequences of surfaces with boundary of fixed length and with arbitrarily large Steklov spectral gap are given. In particular, we obtain such a sequence for surfaces with connected boundary. The applications are based on the construction of graph-like surfaces which are obtained from sequences of graphs with good expansion properties.