Spectrum of the drift Laplacian on Ricci expanders

TL;DR

This paper investigates the spectrum of the drift Laplacian on Ricci expanders, establishing conditions for discreteness, extending asymptotic results, and computing eigenvalues for specific classes of expanders.

Contribution

It demonstrates the discreteness of the spectrum under proper potential functions, extends asymptotic behavior results, and calculates eigenvalues for rigid expanders and shrinkers.

Findings

Spectrum is discrete when the potential function is proper.

Discreteness cannot be assumed without properness.

Eigenvalues are explicitly computed for rigid expanders and shrinkers.

Abstract

In this paper, we study the spectrum of the drift Laplacian on Ricci expanders. We show that the spectrum is discrete when the potential function is proper, and we show that the hypothesis on the properness of the potential function cannot be removed. We also extend previous results concerning the asymptotic behavior of the potential function on Ricci expanders. This allows us to conclude that the drift Laplacian has discrete spectrum on Ricci expanders whose Ricci curvature is bounded below by a suitable constant, possibly negative. Further, we compute all the eigenvalues of the drift Laplacian on rigid expanders and rigid shrinkers. Lastly, we investigate the second eigenvalue of the drift Laplacian on rigid Ricci expanders whose Einstein factor is a closed hyperbolic Riemann surface.