Some local eigenvalue estimates involving curvatures

Abstract

We first establish a local Faber-Krahn isoperimetric comparison in terms of scalar curvature pinching. Secondly we derive estimates of Cheeger constants related to the Dirichlet and Neumann problems via the (relative) isoperimetric profiles which allow us to obtain, in particular, lower bounds for first non-zero eigenvalues of the problem of Dirichlet and Neumann. These estimates involve scalar curvature and mean curvature respectively. Mathematics Subject Classification (2000)15A42 · 35B05 · 35R45 · 52A40

Sharp local estimates for the Szegö–Weinberger profile in Riemannian manifolds

“…This follows from the local expansion of µ 2 in small geodesic balls in these manifolds, see Remark 3.3(ii) below. Corollary 1.2 should be seen in comparison with the results in [6,7,9] concerning the isoperimetric profile I M and the Faber-Krahn profile F K M of M. More precisely, set…”

“…Corollary 1.2 should be seen in comparison with the results in [6,7,9] concerning the isoperimetric profile I M and the Faber-Krahn profile F K M of M. More precisely, set…”

“…Inequality (8) was established by Druet [7], and (9) was derived independently by Druet [6] and the first author [9]. The first ingredient in the proof of (9) is the following expansion of λ 1 (B g (y, r), g) when r → 0:…”

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

“…Again, we can contrast this with the Euclidean case, which states that |∂Ω| 0 ≥ nω 1/n n |Ω| n−1 n = nω n (r 0 (v)) n−1 . Using these later forms of the isoperimetric inequality for small volumes, Druet [11] and Fall [12] proved a Faber-Krahn theorem, and in fact obtained stability estimates. More precisely, they showed that if Sect(g) ≤ −κ 2 and |Ω| g is small, then λ(Ω) ≥ λ(B r ), where B r is the geodesic ball in the model space as before.…”

Profile expansion for the first nontrivial Steklov eigenvalue in Riemannian manifolds

“…To close the introduction, we mention the earlier work in [5,9] on the small volume expansion for the Faber-Krahn profile, which is related to the minimization of the first Dirichlet eigenvalue λ 1 (Ω, g) of −∆ g among subdomains Ω of fixed volume. One important difference between λ 1 (Ω, g) and ν 2 (Ω, g) is the degeneracy of ν 2 in the case of the unit ball and possibly also in the case of maximizing domains on Riemannian manifolds.…”