The Steklov spectrum and coarse discretizations of manifolds with boundary

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.

Full Text (PDF format)

Received 7 January 2018

Accepted 11 January 2019

Published 5 November 2019