  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.
