An Eigenvalue Problem on Negatively Curved Manifolds

Abstract

Consider the eigenvalue problem: Δgu - λKu = 0 \quad in D where D is the unit disc of the complex plane, g is a complete metric conformal to the Poincaré metric on D, and K is the Gaussian curvature. It is shown that if λ > \frac{1}{2} (λ > \frac{1}{4}in the case of K ≤ 0), then the above problem has no positive solutions.