Vanishing of the fundamental gap for (horo) convex domains in hyperbolic space

Xuan Nguyen

For the Laplace operator with Dirichlet boundary conditions on convex domains in $H^n, n ≥ 2$, we prove that the product of the fundamental gap with the square of the diameter can be arbitrarily small for domains of any diameter. This property distinguishes hyperbolic spaces from Euclidean and spherical ones, where the quantity is bounded below by $3 \pi^2$. We finish by talking about horoconvex domains.

http://scgp.stonybrook.edu/video_portal/video.php?id=5349

Simons- Workshop: Forty Years of Ricci Flow: The Geometric-Flow Revolution in Global Differential Geometry