The vanishing of the fundamental gap of convex domains in ${\mathbb{H}}^n$

Abstract

For the Laplace operator with Dirichlet boundary conditions on convex domains in ${\mathbb{H}}^n$, $n\ge 2$, we prove that the product of the fundamental gap with the square of the diameter can be arbitrarily small for domains of any diameter.

20 pages, 6 figures, comments are welcome