Small eigenvalues of closed surfaces

Abstract

We show that the Laplacian of a Riemannian metric on a closed surface S with Euler characteristic χ(S) < 0 has at most −χ(S) small eigenvalues.

Small eigenvalues of surfaces: old and new

“…(Cf. Lemma 2.5 in [1]). Note that it may happen that one of Y + ϕ (ε) or Y − ϕ (ε) is empty; for example, if ϕ is a positive constant, then Y − ϕ (ε) = ∅.…”

“…In a next step, we show that X ϕ (ε) is not empty (cf. Lemma 2.6 in [1]). Since we are working with approximate nodal sets, the argument from the proof of Lemma 5.8 only applies in the case where the Rayleight quotient R(ϕ) < Λ(S) and shows that X ϕ (ε) is not empty, for all sufficiently small ε > 0.…”

“…The ideas in Sévennec's approach proved to be fruitful in the work of Otal [18], Otal-Rosas [19], and our work in [1,2]. Sévennec started by proving a Borsuk-Ulam type theorem (see Lemmata 7 and 8 in [25]) which has the following consequence.…”

“…The starting point of our joint work is this last question on the real analyticity of the Riemannian metric. We could show that the assumption can indeed be removed in the case of closed surfaces with negative Euler characteristic [1,2016]. Later we could show the inequality even in the general case of complete Riemannian metrics on surfaces S of finite type (in the sense of Section 2) [2,2017].…”

“…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.…”