Spectrum of the Lichnerowicz Laplacian on asymptotically hyperbolic surfaces

TL;DR

This paper investigates the spectrum of the Lichnerowicz Laplacian on asymptotically hyperbolic surfaces, revealing the essential spectrum and conditions for eigenvalues, with implications for geometric analysis.

Contribution

It characterizes the essential spectrum of the Lichnerowicz Laplacian on asymptotically hyperbolic surfaces and identifies conditions for the presence of eigenvalues.

Findings

Essential spectrum contains [1/4, +∞)

-2 and 0 are infinite-dimensional eigenvalues under constant scalar curvature

Spectrum is discrete under certain inequalities

Abstract

We show that, on any asymptotically hyperbolic surface, the essential spectrum of the Lichnerowicz Laplacian $\Delta_L$ contains the ray $[{1/4},+\infty[$. If moreover the scalar curvature is constant then -2 and 0 are infinite dimensional eigenvalues. If, in addition, the inequality $<\Delta u, u>_{L^2}\geq \frac14||u||^2_{L^2}$ holds for all smooth compactly supported function $u$, then there is no other value in the spectrum.