A second eigenvalue bound for the Dirichlet Laplacian in hyperbolic space

Abstract

Abstract. Let Ω be some domain in the hyperbolic space H n (with n ≥ 2) and S 1 the geodesic ball that has the same first Dirichlet eigenvalue as Ω. We prove the Payne-Pólya-Weinberger conjecture for H n , i.e., that the second Dirichlet eigenvalue on Ω is smaller or equal than the second Dirichlet eigenvalue on S 1 .We also prove that the ratio of the first two eigenvalues on geodesic balls is a decreasing function of the radius.

Monotonicity of the first Dirichlet eigenvalue of the Laplacian on manifolds of non-positive curvature

“…The dependence on the eigenvalue in the hyperbolic case is more challenging to understand, because the eigenfunctions on geodesic balls do not scale in curved setting (see, for instance, Section 3 of [2]). …”

Eigenvalue pinching on convex domains in space forms

“…It is shown in [4] that λ 2 /λ 1 is a decreasing function of the radius of hyperbolic balls and that the PPW is false in H n . This theorem can be seen as a generalized PPW inequality on space forms.…”

“…We will prove Theorems 4.8 and 4.9 simultaneously. The scheme of the proof is the same as in [1,2,4], so we will mainly focus on the extra arguments needed in our setting. The first step of the proof is the following proposition.…”

“…The next step consists in choosing n suitable test functions. For that purpose, we need the following lemma which extends a result of [1,2,4] (the proof is postponed to Section 7). …”

“…This lemma is essentially proven in [1,2,4] for u with compact support (which includes the case δ = 1), but we will apply it for u an eigenfunction of an unbounded domain Ω and so we have to extend it in the case δ = 0, −1.…”