Dirichlet eigenvalue asymptotics on conformally compact Einstein manifolds

Abstract

We study the first Dirichlet eigenvalues of large geodesic balls and fixed-width boundary layers on conformally compact Einstein manifolds of dimension $n+1.$ For large balls, we determine the exponential rate of convergence when the spectral bottom is less than $\frac{n^2}{4},$ i.e. the bottom of the continuous spectrum. When it equals the bottom of the continuous spectrum, the second coefficient has two possible values, according to whether the continued resolvent is regular or has a pole there. We also give examples with negative Yamabe conformal infinity realizing both threshold alternatives and construct metrics with a Ricci lower bound that have other second coefficients. Under nonnegative Yamabe conformal infinity, we also find the exact second coefficient for the nonlinear $p$-Laplacian on large balls. For layers defined by a geodesic boundary defining function, we compute the first correction and show that its sign characterizes nonnegative Yamabe conformal infinity.