Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Fundamental tones of clamped plates in nonpositively curved spaces   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.

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The categories of this paper are: cs
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The categories of this paper are: physics
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The categories of this paper are: cmp-lg
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The categories of this paper are: plasm-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: solv-int
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The categories of this paper are: q-fin
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The categories of this paper are: patt-sol
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The categories of this paper are: quant-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-th
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The categories of this paper are: hep-lat
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The categories of this paper are: gr-qc
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The categories of this paper are: mtrl-th
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The categories of this paper are: ao-sci
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The categories of this paper are: atom-ph