On the Steklov spectrum of covering spaces and total spaces

TL;DR

This paper introduces a Dirichlet-to-Neumann map on Riemannian manifolds with boundary, analyzing its spectrum on covering and total spaces of principal bundles, revealing new spectral properties in geometric analysis.

Contribution

It constructs a natural Dirichlet-to-Neumann map with positive bottom spectrum and studies its spectral behavior on covering and bundle spaces, advancing geometric spectral theory.

Findings

Existence of a Dirichlet-to-Neumann map with positive bottom spectrum

Spectral analysis of the map on covering spaces

Spectral properties on total spaces of principal bundles

Abstract

We show the existence of a natural Dirichlet-to-Neumann map on Riemannian manifolds with boundary and bounded geometry, such that the bottom of the Dirichlet spectrum is positive. This map regarded as a densely defined operator in the $L^2$-space of the boundary admits Friedrichs extension. We focus on the spectrum of this operator on covering spaces and total spaces of Riemannian principal bundles over compact manifolds.