Sharp spectral gap estimates for higher-order operators on Cartan-Hadamard manifolds

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.
Comment: 13 pages; to appear in Communications in Contemporary Mathematics (https://doi.org/10.1142/S0219199724500135)