Pure and Applied Mathematics Quarterly

Volume 14 (2018)

Number 2

The Steklov spectrum and coarse discretizations of manifolds with boundary

Pages: 357 – 392

DOI: https://dx.doi.org/10.4310/PAMQ.2018.v14.n2.a3


Bruno Colbois (Institut de Mathématiques, Université de Neuchâtel, Switzerland)

Alexandre Girouard (Département de mathématiques et de statistique, Université Laval, Québec, QC, Canada)

Binoy Raveendran (Indian Institute of Technology, Kanpur, India)


Given $\kappa , r_0 \gt 0$ and $n \in \mathbb{N}$, we consider the class $\mathcal{M} = \mathcal{M} (\kappa , r_0, n)$ of compact $n$-dimensional Riemannian manifolds with cylindrical boundary, Ricci curvature bounded below by $- (n - 1) \kappa$ and injectivity radius bounded below by $r_0$ away from the boundary. For a manifold $M \in \mathcal{M}$ we introduce a notion of discretization, leading to a graph with boundary which is roughly isometric to $M$, with constants depending only on $\kappa , r_0, n$. In this context, we prove a uniform spectral comparison inequality between the Steklov eigenvalues of a manifold $M \in \mathcal{M}$ 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 $\sigma_2 - \sigma_1$ 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.

Received 7 January 2018

Accepted 11 January 2019

Published 5 November 2019