The Steklov spectrum and coarse discretizations of manifolds with boundary

TL;DR

This paper establishes a spectral comparison between the Steklov eigenvalues of certain Riemannian manifolds with boundary and their discretized graph models, enabling new constructions of surfaces with large spectral gaps.

Contribution

It introduces a discretization method for manifolds with boundary and proves a uniform spectral comparison inequality, linking manifold spectra to graph spectra.

Findings

Spectral comparison inequality between manifold and discretization

Construction of surfaces with fixed boundary length and large spectral gap

Application of graph expansion properties to geometric analysis

Abstract

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.