  The goal of this paper is to provide sharp spectral gap estimates for
problems involving higher-order operators (including both the clamped and
buckling plate problems) on Cartan-Hadamard manifolds. The proofs are
symmetrization-free -- thus no sharp isoperimetric inequality is needed --
based on two general, yet elementary functional inequalities. The spectral gap
estimate for clamped plates solves a sharp asymptotic problem from Cheng and
Yang [Proc. Amer. Math. Soc., 2011] concerning the behavior of higher-order
eigenvalues on hyperbolic spaces, and answers a question raised in Krist\'aly
[Adv. Math., 2020] on the validity of such sharp estimates in high-dimensional
Cartan-Hadamard manifolds. As a byproduct of the general functional
inequalities, various Rellich inequalities are established in the same
geometric setting.
