A lower bound to the spectral threshold in curved tubes

TL;DR

This paper establishes a lower bound for the spectral threshold of the Laplacian in curved tubes with arbitrary cross-sections, linking it to the eigenvalues of a related Dirichlet problem based on the tube's geometry.

Contribution

It provides a novel geometric lower bound for the spectral threshold in curved tubes of arbitrary dimension and cross-section, extending previous results to more general settings.

Findings

Spectral threshold is bounded below by the lowest eigenvalue of a related Dirichlet problem.

The bound depends on the geometry of the tube and the eigenvalues of the cross-section.

Results apply to tubes in Euclidean spaces of arbitrary dimension.

Abstract

We consider the Laplacian in curved tubes of arbitrary cross-section rotating together with the Frenet frame along curves in Euclidean spaces of arbitrary dimension, subject to Dirichlet boundary conditions on the cylindrical surface and Neumann conditions at the ends of the tube. We prove that the spectral threshold of the Laplacian is estimated from below by the lowest eigenvalue of the Dirichlet Laplacian in a torus determined by the geometry of the tube.