New features of the first eigenvalue on negatively curved spaces

The paper is devoted to the study of fine properties of the first eigenvalue on negatively curved spaces. First, depending on the parity of the space dimension, we provide asymptotically sharp harmonic-type expansions of the first eigenvalue for large geodesic balls in the model $n$-dimensional hyperbolic space, complementing the results of Borisov and Freitas (\textit{Comm. Anal. Geom.} 25: 507--544, 2017). We then give a synthetic proof of Cheng's sharp eigenvalue comparison theorem in metric measure spaces satisfying a 'negatively curved' Bishop-Gromov-type volume monotonicity hypothesis. As a byproduct, we provide an example of simply connected, non-compact Finsler manifold with constant negative flag curvature whose first eigenvalue is zero; this result is in a sharp contrast with its celebrated Riemannian counterpart due to McKean (\textit{J. Differential Geom.} 4: 359--366, 1970). Our proofs are based on specific properties of the Gaussian hypergeometric function combined with intrinsic aspects of the negatively curved smooth/non-smooth spaces.