论文标题
负曲率限制了凸域的基本差距
Negative curvature constricts the fundamental gap of convex domains
论文作者
论文摘要
我们考虑了Riemannian歧管$（M^n，g）$在凸域上具有差异域边界条件的Laplace-Beltrami操作员，并证明，只要$ M^n $具有负截面曲率的单个切线平面，就可以任意地使用直径平方的基本间隙的产物。特别是，对于欧几里得空间的小变形而言，基本的间隙猜想强烈失败，这引入了任何负曲率。我们还表明，当曲率被负捏时，可以构造直径到歧管直径的任何直径的域。证明是根据Bourni等人的论点进行的。 al。 （Annales HenriPoincaré2022），它建立了双曲线空间中凸形域的类似结果，但需要几种新成分。
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 Poincaré 2022), which established the analogous result for convex domains in hyperbolic space, but requires several new ingredients.
