Small eigenvalues of surfaces: old and new
Let \((S_\gamma, g)\) be an orientable closed surface of genus \(\gamma \geq 2\) and with Riemannian metric \(g\). Within the conformal class of \(g\), there is a metric with constant curvature \(K\); when \(K=-1\), the metric is \textit{hyperbolic}. The eigenvalues of the Laplace operator on \((S_\gamma, g)\) are well-studied, and the present article focuses on the so-called \textit{small} eigenvalues, which are those less than \(1/4\) (the smallest eigenvalue for the hyperbolic plane). Work of \textit{B. Randol} [Bull. Am. Math. Soc. 80, 996--1000 (1974; Zbl 0317.30017)] and \textit{P. Buser} [Comment. Math. Helv. 52, 25--34 (1977; Zbl 0348.53027)] showed that one may construct hyperbolic metrics on \(S_\gamma\) with arbitrarily many small eigenvalues; one theme of the present article is to generalize this work to the setting of surfaces of \emph{finite type}. A connected surface \(S\) of finite type has a unique representation as a closed orientable surface \(S_\gamma\) of genus \(\gamma \geq 0\) with \(p \geq 0\) punctures, \(q \geq 0\) holes, and \(0 \leq r \leq 2\) embedded Möbius bands. Another theme in this paper is the question of whether \(\lambda_{2\gamma - 2}\) is small. \textit{J.-P. Otal} and \textit{E. Rosas} [Duke Math. J. 150, No. 1, 101--115 (2009; Zbl 1179.30041)] showed that there is a strict lower bound on \(\lambda_{2\gamma-2}\) given by the bottom of the spectrum of the universal covering surface of \(S_\gamma\), endowed with the lifted metric. They assumed that the metric on \(S_\gamma\) was real analytic and negatively curved. In a series of papers, the present authors have examined various generalizations of this result; these include removing the assumption that the metric be real analytic (under certain hypotheses), extending the result to surfaces of finite type, and giving an improved lower bound in terms of the so-called \textit{analytic systole}. The present paper explains the main ideas behind these generalizations while leaving aside the technical details, making it a reasonable introduction to the results shown in [\textit{W. Ballmann} et al., J. Differ. Geom. 103, No. 1, 1--13 (2016; Zbl 1341.53066); Compos. Math. 153, No. 8, 1747--1768 (2017; Zbl 1401.35216); Geom. Funct. Anal. 27, No. 5, 1070--1105 (2017; Zbl 1390.30052)].
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