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. (in: Annales Henri Poincaré, Springer, 2022), which established the analogous result for convex domains in hyperbolic space, but requires several new ingredients.

我们考虑黎曼流形\((M^n,g)\)中凸域上具有狄利克雷边界条件的拉普拉斯-贝尔特拉米算子，并证明只要\ (M^n\)甚至具有负截面曲率的单个切平面。特别是，对于引入任何负曲率的欧几里得空间的小变形，基本间隙猜想强烈失败。我们还表明，当曲率受到负收缩时，可以构造任意直径的域，直至流形的直径。该证明改编自 Bourni 等人的论点。（见：Annales Henri Poincaré，Springer，2022），它建立了双曲空间中凸域的类似结果，但需要一些新的成分。