Asymptotic behavior of the first Dirichlet eigenvalue of AHE manifolds (Q6614902)

The paper contains two main results on the rate at which the first Dirichlet eigenvalue of the Laplace-Beltrami operator on geodesic balls of certain manifolds with conformal infinity decreases as the radius goes to infinity. In particular, it is shown that if the conformal infinity of an asymptotically hyperbolic Einstein manifold is of nonnegative Yamabe type, then the two term asymptotic of the eigenvalues is the same as that in hyperbolic space:\N\[\N\forall p \in (M^{n+1}g):\ \ \lambda_1(B(p,R))=\frac{n^2}{4} + \frac{\pi^2}{R^2} + O (R^{-3}),\ \ {\hbox{for}}\ \ R \ {\hbox{sufficiently large}}.\N\]