An Eigenvalue Problem on Negatively Curved Manifolds (Journal of Partial Differential Equations, 2020)
Consider the eigenvalue problem: &#916gu - &#955Ku = 0 \quad in D where D is the unit disc of the complex plane, g is a complete metric conformal to the Poincar&#233 metric on D, and K is the Gaussian curvature. It is shown that if &#955 > \frac{1}{2} (&#955 > \frac{1}{4}in the case of K &#8804 0), then the above problem has no positive solutions.
Keywords: Eigenvalue problem; complete manifolds; negative curvature