Small eigenvalues of surfaces of finite type

Abstract

Abstract. Extending our previous work on eigenvalues of closed surfaces and work of Otal and Rosas, we show that a complete Riemannian surface S of finite type and Euler characteristic χ(S) < 0 has at most −χ(S) small eigenvalues.

On the Spectrum of Differential Operators Under Riemannian Coverings

“…Since η is smooth and compactly supported in B(x, r), there exists C 2 > 0, such that in any of these balls, denoting by (η (1) , . .…”

“…Therefore, we obtain uniform estimates up to order d for χ, which are independent from P Clearly, for z ∈ supp(χθ), we have that z ∈ B(y, r), for some y ∈ p −1 (x), and in particular, z is contained in a ball of the system. With respect to the corresponding coordinate system and trivialization, denoting by (θ (1) , . .…”

Small eigenvalues of surfaces: old and new

“…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.…”

“…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) = ∅. Similar to the argument in Section 5.3, we restrict our attention to those components of Y ϕ (ε) and Y ± ϕ (ε) having negative Euler characteristic and write X ϕ (ε) and X ± ϕ (ε) for the union of these components, respectively (now following the notation in [2]).…”

“…In particular, we obtain a quantitative generalization of Theorem 1.1 by elementary geometric arguments. In Section 5, we discuss the ideas of Sévennec, Otal, and Otal-Rosas which were important in our work in [1,2].…”