Absence of Principal Eigenvalues for Higher Rank Locally Symmetric Spaces

Abstract

AbstractGiven a geometrically finite hyperbolic surface of infinite volume it is a classical result of Patterson that the positive Laplace–Beltrami operator has no $$L^2$$ L 2 -eigenvalues $$\ge 1/4$$ ≥ 1 / 4 . In this article we prove a generalization of this result for the joint $$L^2$$ L 2 -eigenvalues of the algebra of commuting differential operators on Riemannian locally symmetric spaces $$\Gamma \backslash G/K$$ Γ \ G / K of higher rank. We derive dynamical assumptions on the $$\Gamma $$ Γ -action on the geodesic and the Satake compactifications which imply the absence of the corresponding principal eigenvalues. A large class of examples fulfilling these assumptions are the non-compact quotients by Anosov subgroups.