New features of the first eigenvalue on negatively curved spaces

Abstract

<jats:title>Abstract</jats:title><jats:p>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 <jats:italic>n</jats:italic>-dimensional hyperbolic space, complementing the results of Borisov and Freitas (2017), Hurtado, Markvorsen and Palmer (2016) and Savo (2008); in odd dimensions, such eigenvalues appear as roots of an inductively constructed transcendental equation. We then give a synthetic proof of Cheng’s sharp eigenvalue comparison theorem in metric measure spaces satisfying a 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 (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.</jats:p>

Fundamental tones of clamped plates in nonpositively curved spaces

“…Incidentally, it turns out that for n = 2 (ν = 0), the function t → G − (0, λ, t) := F 1−Λ − 2 , 1+Λ − 2 ; 1; −t appears as the extremal in the second-order Rayleigh problem (for membranes) on the geodesic ball B κ (L 0 ) with the initial condition F 1−Λ − 2 , 1+Λ − 2 ; 1; − L0 = 0 where L0 = sinh( κL 0 2 ) 2 , see e.g. Kristály [26], while the first eigenvalue γ g (B κ (L 0 )) corresponding to (1.6) on B κ (L 0 ) is precisely g 0,1 ( L0 ). Therefore, by Chavel [6,p.318] one has that…”

Three isometrically equivalent models of the Finsler-Poincaré disk

“…Most importantly, it implies that all the metric related properties which are enjoyed by one particular model can be easily proved to hold on the other two manifolds as well. In particular, based on Kristály [12], we find arXiv:2204.03052v1 [math.DG] 6 Apr 2022 that the first Dirichlet eigenvalue λ F associated to the Finsler-Laplace operator −∆ F is zero in the case of both Finsler-Poincaré models (P) and (H). This provides new examples of simply connected, non-compact Finsler manifolds with constant negative flag curvature having zero first eigenvalue, which is an unexpected result considering its Riemannian counterpart proven by McKean [15].…”

“…* 2 (x, Du(x))dv F (x) M u 2 (x)dv F (x), where H 1 0,F (M ) is the closure of C ∞ 0 (M ) with respect to the normu H 1 0,F = M (x, Du(x))dv F (x) + M u 2 (x)dv F (x)1 see Ge and Shen[10], Ohta and Sturm[16].According to Kristály[12, Theorem 1.3], in case of the Finslerian Funk model (D, F F ), we have thatλ 1,F F (D) = 0.Combining this with the isometries proven in Theorems 1 and 2, we obtain the following result: In case of the Finsler-Poincaré disk (D, F P ) and the Finslerian upper half plane (H, F H ), we have λ 1,F P (D) = λ 1,F H (H) = 0.…”

Optimal eigenvalue estimates for the Robin Laplacian on Riemannian manifolds

“…When σ = +∞ (Dirichlet problem) the lower bound in (9) was improved in [2] and the two-term expansion has been refined in Theorem 1.1 of [16].…”