We study Lord Rayleigh's problem for clamped plates on an arbitrary $n$-dimensional $(n\geq 2)$ Cartan-Hadamard manifold $(M,g)$ with sectional curvature $\textbf{K}\leq -\kappa^2$ for some $\kappa\geq 0.$ We first prove a McKean-type spectral gap estimate, i.e. the fundamental tone of any domain in $(M,g)$ is universally bounded from below by $\frac{(n-1)^4}{16}\kappa^4$ whenever the $\kappa$-Cartan-Hadamard conjecture holds on $(M,g)$, e.g. in 2- and 3-dimensions due to Bol (1941) and Kleiner (1992), respectively. In 2- and 3-dimensions we prove sharp isoperimetric inequalities for sufficiently small clamped plates, i.e. the fundamental tone of any domain in $(M,g)$ of volume $v>0$ is not less than the corresponding fundamental tone of a geodesic ball of the same volume $v$ in the space of constant curvature $-\kappa^2$ provided that $v\leq c_n/\kappa^n$ with $c_2\approx 21.031$ and $c_3\approx 1.721$, respectively. In particular, Rayleigh's problem in Euclidean spaces resolved by Nadirashvili (1992) and Ashbaugh and Benguria (1995) appears as a limiting case in our setting (i.e. $\textbf{K}\equiv\kappa=0$). The sharpness of our results requires the validity of the $\kappa$-Cartan-Hadamard conjecture (i.e. sharp isoperimetric inequality on $(M,g)$) and peculiar properties of the Gaussian hypergeometric function, both valid only in dimensions 2 and 3; nevertheless, some nonoptimal estimates of the fundamental tone of arbitrary clamped plates are also provided in high-dimensions. As an application, by using the sharp isoperimetric inequality for small clamped hyperbolic discs, we give necessarily and sufficient conditions for the existence of a nontrivial solution to an elliptic PDE involving the biharmonic Laplace-Beltrami operator.

中文翻译：

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非正弯曲空间中夹板的基调

我们研究了在任意 $n$ 维 $(n\geq 2)$ Cartan-Hadamard 流形 $(M,g)$ 上的夹板的瑞利勋爵问题，截面曲率 $\textbf{K}\leq -\kappa^ 2$ 对于某些 $\kappa\geq 0.$ 我们首先证明了 McKean 类型的谱间隙估计，即 $(M,g)$ 中任何域的基调都普遍受 $\frac{(n -1)^4}{16}\kappa^4$ 只要 $\kappa$-Cartan-Hadamard 猜想在 $(M,g)$ 上成立，例如在 2 维和 3 维中，由于 Bol (1941) 和克莱纳 (1992)，分别。在 2 维和 3 维中，我们证明了足够小的夹紧板的尖锐等周不等式，即体积 $v> 中 $(M,g)$ 中的任何域的基调 0$不小于相同体积$v$在等曲率空间$-\kappa^2$的测地球对应的基音，条件是$v\leq c_n/\kappa^n$与$c_2 \approx 21.031$ 和 $c_3\approx 1.721$，分别。特别是，由 Nadirashvili (1992) 和 Ashbaugh 和 Benguria (1995) 解决的欧几里得空间中的瑞利问题在我们的设置中作为一个极限情况出现（即 $\textbf{K}\equiv\kappa=0$）。我们的结果的清晰性需要 $\kappa$-Cartan-Hadamard 猜想（即 $(M,g)$ 上的尖锐等周不等式）和高斯超几何函数的特殊性质的有效性，两者都仅在维度 2 和 3 中有效; 尽管如此，在高维中也提供了对任意夹紧板的基音的一些非最佳估计。作为应用程序，