Spectrum of the Lichnerowicz Laplacian on asymptotically hyperbolic surfaces

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 $<\Delta u,u{>}_{L^2}\ge \frac{1}{4}||u|{|}_{L^2}^2$ holds for all smooth compactly supported function $u$, then there is no other value in the spectrum.

13 pages

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