Negative curvature constricts the fundamental gap of convex domains

2022/11/11 by Gabriel Khan, Khan, Gabriel, Xuan Hien Nguyen +1 · 2 citations
Mathematics · #58J50 35P15 #Algebraic and Geometric Analysis #Analysis of PDEs (math.AP) #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2211.06404

openalex publication_date 2022/11/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the Laplace-Beltrami operator with Dirichlet boundary conditions on convex domains in a Riemannian manifold (Mⁿ,g), and prove that the product of the fundamental gap with the square of the diameter can be arbitrarily small whenever Mⁿ 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.

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