The spectrum of an asymptotically hyperbolic Einstein manifold

TL;DR

This paper investigates the spectrum of the scalar Laplacian on asymptotically hyperbolic Einstein manifolds, showing that non-negative Yamabe invariant on the boundary prevents discrete eigenvalues below the essential spectrum.

Contribution

It establishes a link between the boundary conformal geometry's Yamabe invariant and the absence of discrete eigenvalues in the spectrum.

Findings

Essential spectrum is [n^2/4, ∞) with no embedded eigenvalues.

Non-negative Yamabe invariant implies no discrete eigenvalues below the essential spectrum.

Generalizes previous results for hyperbolic manifolds.

Abstract

This paper relates the spectrum of the scalar Laplacian of an asymptotically hyperbolic Einstein metric to the conformal geometry of its ``ideal boundary'' at infinity. It follows from work of R. Mazzeo that the essential spectrum of such a metric on an $(n+1)$-dimensional manifold is the ray $[n^2/4,\infty)$, with no embedded eigenvalues; however, in general there may be discrete eigenvalues below the continuous spectrum. The main result of this paper is that, if the Yamabe invariant of the conformal structure on the boundary is non-negative, then there are no such eigenvalues. This generalizes results of R. Schoen, S.-T. Yau, and D. Sullivan for the case of hyperbolic manifolds.