Negative curvature constricts the fundamental gap of convex domains

TL;DR

This paper demonstrates that negative curvature in a Riemannian manifold can drastically reduce the fundamental gap of convex domains, invalidating the gap conjecture in such settings.

Contribution

It shows that negative curvature can make the fundamental gap arbitrarily small, even for convex domains, challenging previous conjectures and extending results to more general manifolds.

Findings

Fundamental gap times diameter squared can be arbitrarily small with negative curvature.

Negative curvature invalidates the fundamental gap conjecture in convex domains.

Construction of convex domains with small gaps in negatively curved manifolds.

Abstract

We consider the Laplace-Beltrami operator with Dirichlet boundary conditions on convex domains in a Riemannian manifold $(M^n,g)$, and prove that the product of the fundamental gap with the square of the diameter can be arbitrarily small whenever $M^n$ has even a single tangent plane of negative sectional curvature. In particular, the fundamental gap conjecture strongly fails for small deformations of Euclidean space which introduce any negative curvature. We also show that when the curvature is negatively pinched, it is possible to construct such domains of any diameter up to the diameter of the manifold. The proof is adapted from the argument of Bourni et. al. (Annales Henri Poincar\'e 2022), which established the analogous result for convex domains in hyperbolic space, but requires several new ingredients.