Riemann surfaces with large first eigenvalue
Riemann surfaces with large first eigenvalue
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具有大第一特征值的黎曼曲面
DOI：
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发表时间：
2001
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通讯作者：
Eran Makover
中科院分区：
文献类型：
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作者：
R. Brooks;Eran Makover
在本文中，我们考虑构造具有第一特征值λ 1 (S)较大的任意亏格的紧黎曼曲面S的问题。在本文中，通过短语“黎曼曲面”，我们理解具有恒定曲率 -1 的完全度量的定向曲面。为了精确地陈述我们的结果，让我们设置 Λ(g) = λ 1 (S g) 的最大值，其中 S g 的范围在属 g 的紧致黎曼曲面上。 Buser [Bu1] 于 1978 年提出了 Λ(g) 对于大 g 的行为问题，他猜想 lim g→∞ Λ(g) = 0。随后，他使用数论中的重型机械在 [Bu2] 中证明 lim su g→∞ Λ(g) ≥ 3/16。许多这种重型机械后来在 [BBD] 中被移除，但出于数论考虑，[Bu2] 和 [BBD] 仅在某些属中构造了大 λ 1 的表面。本文的主要结果，即下面的定理 1.4，给出了一种修改具有大第一特征值的黎曼曲面族以获得更大的族的方法，该族保留了第一特征值大的性质。特别地，以这种方式构建的族可以包括所有属的表面，即使第一族仅包含非常特殊属的表面。在定理 1.2 中，我们将此构造应用于非紧模曲面 P (k) = H 2 /Γ k ，如下所述。 P(k)的亏格近似为k的三次方，而尖点的数量近似为k的二次方；详细信息请参见引理 6.1。对于 C = inf k λ 1 (P (k))，我们有定理 1.2 lim inf g→∞ Λ(g) ≥ C。根据 Selberg 的著名定理 [Sel]，C ≥ 3/16。这已被 Luo-Rudnick-Sarnak [LRS] 改进为 C ≥ 171/784。塞尔伯格在 [Sel] 中猜想 C = 1/4。我们注意到上限 lim su g→∞ Λ(g) ≤ 1/4 1 已经在 [Bu1] 中观察到。在第 §1 节中，我们还给出了带有尖点的黎曼曲面结果的一个版本。这篇论文的结果已经在[BM2]中公布。 [BM] 中的作者给出了具有大 λ 1 的黎曼曲面的一种完全不同的构造，它不依赖于任何数论，但它为 lim inf g→∞ Λ(g) 产生了更小的且非显式的界限。致谢：……
In this paper, we consider the problem of constructing compact Riemann surfaces S of arbitrary genus with the property that the first eigenvalue λ 1 (S) is large. Throughout this paper, by the phrase " Riemann surface " we understand an oriented surface with a complete metric of constant curvature −1. To state our results precisely, let us set Λ(g) = the maximum value of λ 1 (S g), where S g ranges over compact Riemann surfaces of genus g. The question of the behavior of Λ(g) for large g was raised by Buser [Bu1] in 1978, where he conjectured that lim g→∞ Λ(g) = 0. He subsequently showed in [Bu2] that lim sup g→∞ Λ(g) ≥ 3/16, using heavy machinery from number theory. Much of this heavy machinery was later removed in [BBD], but both [Bu2] and [BBD] construct surfaces of large λ 1 only in certain genera arising from number-theoretic considerations. The main result of this paper, Theorem 1.4 below, gives a method for modifying a family of Riemann surfaces with large first eigenvalue to obtain a much larger family which retains the property that the first eigenvalue is large. In particular, the family constructed in this way may include surfaces of all genera, even though the first family contains surfaces of only very special genera. In Theorem 1.2, we apply this construction to the noncompact modular surfaces P (k) = H 2 /Γ k , to be described below. The genus of P (k) is approximately cubic in k, while the number of cusps is approximately quadratic in k; see Lemma 6.1 for details. For C = inf k λ 1 (P (k)), we have Theorem 1.2 lim inf g→∞ Λ(g) ≥ C. According to a famous theorem of Selberg [Sel], C ≥ 3/16. This has been improved by Luo-Rudnick-Sarnak [LRS] to C ≥ 171/784. Selberg conjectured in [Sel] that C = 1/4. We remark that the upper bound lim sup g→∞ Λ(g) ≤ 1/4 1 was already observed in [Bu1]. In Section §1, we also give a version of this result for Riemann surfaces with cusps. The results of this paper have been announced in [BM2]. A quite different construction of Riemann surfaces with large λ 1 , which relies on no number theory whatsoever, but which produces a smaller and non-explicit bound for lim inf g→∞ Λ(g), is given by the authors in [BM]. Acknowledgements: …