论文标题
雷利勋爵（Lord Rayleigh）的猜想，用于在弯曲的空间
Lord Rayleigh's Conjecture for Vibrating Clamped Plates in Positively Curved Spaces
论文作者
论文摘要
我们肯定地解决了雷利勋爵对在2和3维度中的任何夹板板的构想中的猜想的类似物，以及在3个维度上的足够大的夹板板中的任何夹紧板，这些结果是超过3的。这些结果补充了这些结果。 2020）仅在2和3维度有效的案例，同时还为雷利勋爵的猜想提供了第一个积极答案。这些证明依赖于Ashbaugh-benguria-nadirashvili-talenti nodal分解论点，在Lévy-Gromov等等不平等的情况下，在高斯高几幅功能的精细特性以及高斯和大型小倾斜球形盖的基本色调的尖锐光谱估计上。我们的结果表明，正曲率增强了低维环境和高维度之间的真实差异，这是振动夹板板的理论中默认接受的范式。在极限情况下 - 当RICCI曲率是非负的时 - 我们建立了一个雷利型等等的不平等现象，涉及非紧凑型完整Riemannian歧管的渐近体积比；此外，在2和3维度中，不平等强度是刚性的，即，如果对给定的夹板板的平等性保持相等，则该歧管是欧几里得空间的等值线。
We affirmatively solve the analogue of Lord Rayleigh's conjecture on Riemannian manifolds with positive Ricci curvature for any clamped plates in 2 and 3 dimensions, and for sufficiently large clamped plates in dimensions beyond 3. These results complement those from the flat (M. Ashbaugh & R. Benguria, 1995, and N. Nadirashvili, 1995) and negatively curved (A. Kristály, 2020) cases that are valid only in 2 and 3 dimensions, and at the same time also provide the first positive answer to Lord Rayleigh's conjecture in higher dimensions. The proofs rely on an Ashbaugh-Benguria-Nadirashvili-Talenti nodal-decomposition argument, on the Lévy-Gromov isoperimetric inequality, on fine properties of Gaussian hypergeometric functions and on sharp spectral gap estimates of fundamental tones for both small and large clamped spherical caps. Our results show that positive curvature enhances genuine differences between low- and high-dimensional settings, a tacitly accepted paradigm in the theory of vibrating clamped plates. In the limit case - when the Ricci curvature is non-negative - we establish a Lord Rayleigh-type isoperimetric inequality that involves the asymptotic volume ratio of the non-compact complete Riemannian manifold; moreover, the inequality is strongly rigid in 2 and 3 dimensions, i.e., if equality holds for a given clamped plate then the manifold is isometric to the Euclidean space.
