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 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.

The fundamental gap of horoconvex domains in $\mathbb H^n$

“…To find an upper bound estimate for λ 2 , we will seek in the next section an upper bound for the first eigenvalue of a more general Schrödinger equation and, as such, we will simultaneously obtain an upper bound for λ 1 , slightly improve the one in (5).…”

“…In hyperbolic space, this quantity behaves very differently from the Euclidean and spherical cases. Recently, the authors showed [5] that for any fixed D > 0, there are convex domains with diameter D in H n , n ≥ 2, such that D 2 (λ 2 − λ 1 ) is arbitrarily small. Since convexity does not provide a lower bound, one naturally asks if imposing a stronger notion of convexity, such as horoconvexity, would imply an estimate for D 2 (λ 2 − λ 1 ) from below.…”

“…In the authors' earlier work [5], it was shown that, for any fixed D > 0, one can find a domain Ω for which (λ 2 (Ω) − λ 1 (Ω))D 2 can be made arbitrarily small. The domains Ω ⊂ H n in [5] are convex, but not horoconvex. Their first eigenfunction is not logconcave either.…”

Fundamental gaps of spherical triangles

“…Recently, Dai, He, Seto, Wang, and Wei (in various subsets) [4,6,9] generalized the estimate to convex domains in S n , showing that the same bound holds: λ 2 − λ 1 ≥ 3π 2 /D 2 . Very recently, the second author with coauthors [3] showed the surprising result that there is no lower bound on the fundamental gap of convex domain in the hyperbolic space with arbitrary fixed diameter. This is done by estimating the fundamental gap of some suitable convex thin strips.…”