The eigenvalues of the Laplace operator are important in both mathematics and physics. In this talk we focus on understanding the spectral gap - the difference between the first two eigenvalues.

A series of celebrated results established a beautiful lower bound for this gap on convex domains in Euclidean space. If one assumes Dirichlet boundary conditions, the gap is always at least 3π²/D², where D is the diameter of the domain. In this talk, we discuss how the spectral gap behaves on Riemannian manifolds. In particular, we will see an interesting vanishing of the fundamental gap in negatively curved spaces. In hyperbolic space, one can find convex domains of any diameter whose spectral gap is as small as one likes.

This raises a natural question: if convexity alone fails to guarantee a lower bound on the spectral gap, is there a stronger notion of convexity that ensures a lower bound? Nguyen, Stancu, and Wei conjectured that the answer is yes. We will describe recent work confirming their conjecture and establishing lower bounds on the spectral gap for sufficiently convex regions - called horoconvex - in hyperbolic space.