Pour toute surface hyperbolique de genre g, λ2g−2>1/4

On the Spectra of Hyperbolic Surfaces Without Thin Handles

“…. , f l : Ω → R with disjoint supports and such that Ω f 2 j ≥ c 1 · k 3 , Ω | grad f j | 2 ≤ c 2 · k (constants c 1 , c 2 are absolute). By the geometric version of minimax principle (that is, by upper estimate from Dirichlet-Neumann bracketing), this leads to the desired eigenvalue estimate.…”

“…Proposition 8 below shows that our estimate is sharp in the order. A theorem by Otal and Rosas ( [3]) says that λ 2g−2 > 1/4 for any Ω of genus g. To the other hand, for a given δ > 0, N ∈ N and g = 2, 3, . .…”

Small eigenvalues of surfaces: old and new

“…The development so far is what we refer to as old in our title, and our presentation of it is trimmed towards our needs. The new development starts with the work of Otal and Rosas, who proved the following strengthened version of the Buser-Schmutz conjecture in [19,2009], using ideas from Sévennec [25,2002] and Otal [18,2008]. Theorem 1.4 (Otal-Rosas).…”

“…At the expense of the strictness of the inequality, the assumption can be removed, by the density of the space of real analytic Riemannian metrics inside the space of smooth ones. In [19,Question 2], Otal and Rosas speculate about the possibility of removing the assumption, keeping the strictness of the inequality.…”

“…The rather long Section 6 discusses our results from [1]- [3]. First, in Section 6.1, we describe the arguments needed to extend the ideas from [19] to get the improved bound on the number of small eigenvalues. We then proceed in Section 6.2 and Section 6.3 to the bounds for the analytic systole obtained in [3].…”