The difference between the first two eigenvalues of the Dirichlet Laplacian (the fundamental gap) on convex domains in Euclidean space and, respectively, on the sphere satisfies the same strictly positive lower bound depending on the square of the diameter of the domain, independent of the dimension of the space. In work with collaborators, we found that the fundamental gap in hyperbolic space on convex sets behaves strikingly differently, even when a stronger notion of convexity, namely horoconvexity, is imposed. This is particularly interesting because other features of the eigenvalues behave similarly across all three spaces of constant sectional curvature. We conjectured a lower bound on the fundamental gap in hyperbolic space for horoconvex domains, depending on both the diameter and the dimension. Partial results supporting this conjecture were subsequently obtained by others, and I will also report on these developments.
