On the Lp-Spectrum of Uniformly Elliptic Operators on Riemannian Manifolds

We prove that the Lp spectrum of uniformly elliptic divergence form operators on a complete Riemannian manifold is independent of p ∈ [1, ∞] if the volume of the manifold grows uniformly subexponentially. The latter condition is satisfied if the Ricci curvature is bounded from below by a function K: M → R satisfying lim infx → ∞K(x) ≥ 0, in particular, if the Ricci curvature is nonnegative. On the other hand, if we asume the existence of a (globally defined) volume density which grows exponentially in every direction, then the Lp spectrum depends on p. This condition is satisfied if the manifold is simply connected and if the sectional curvature is bounded from above by a function K: M → R satisfying K ≤ 0 and lim supx → ∞K(x) < 0. For instance, it is satisfied for the hyperbolic space.