ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 5, Pages 76–81 (June 28, 1999) S 1079-6762(99)00064-5

THE FIRST EIGENVALUE OF A RIEMANN SURFACE

ROBERT BROOKS AND ERAN MAKOVER

(Communicated by Walter Neumann)

Abstract. We present a collection of results whose central theme is that the phenomenon of the first eigenvalue of the Laplacian being large is typical for Riemann surfaces. Our main analytic tool is a method for studying how the hyperbolic metric on a Riemann surface behaves under compactification of the surface. We make the notion of picking a Riemann surface at random by modeling this process on the process of picking a random 3-regular graph. With this model, we show that there are positive constants C_{1} and C_{2} independent of the genus, such that with probability at least C_{1}, a randomly picked surface has first eigenvalue at least C_{2}.

In this note, we announce a collection of results ([10, 11, 12]) connected to the behavior of the first eigenvalue λ_{1}(S) of a compact Riemann surface of large genus, endowed with a metric of constant curvature −1. These results have as their common theme that the phenomenon of λ_{1} large is in some sense typical. To make the notion of “typical” precise, we model the process of picking a Riemann surface at random on the process of picking a 3-regular graph at random.

The idea of studying the first eigenvalue of a Riemann surface via the study of eigenvalues of 3-regular graphs comes from the work of Buser [13, 14]. In effect, our approach here is a variation on his idea, where we first study the behavior of λ_{1} on finite-area Riemann surfaces connected to 3-regular graphs, and then see how λ_{1} changes when we compactify the surface.

Our main analytic tool is a method for studying how the hyperbolic metric of a finite-area Riemann surface behaves under such a compactification. This method was introduced in [6], and is based on the Ahlfors-Schwarz Lemma ([1]; see also [5]).

We then have:

Theorem 1 ([10]). For all ϵ, there exists N such that, for g ≥ N, there is a compact Riemann surface S_{g} of genus g satisfying

λ_{1}(S_{g}) ≥ \frac{171}{784} − ϵ.

The number 171/784 comes from the improvement by Luo, Rudnick, and Sarnak [17] of the Selberg 3/16 Theorem [18]. If Selberg’s conjecture were true, we would be able to replace 3/16 by 1/4, the best possible value. More generally, Theorem 1

\begin{matrix}c \\ ◯ \end{matrix}1999 American Mathematical Society

Received by the editors March 25, 1999.

1991 Mathematics Subject Classification. Primary 58G99.

Partially supported by the Israel Science Foundation, founded by the Israel Academy of Arts and Sciences, the Fund for the Promotion of Research at the Technion, and the New York Metropolitan Fund.

contains a method to build from a collection of Riemann surfaces {S_{i}} of large first eigenvalue a larger collection of surfaces whose genera may include all but finitely many genera, whose first eigenvalues satisfy a similar bound.

To state our next results, let F_{n,k} denote the set of k-regular graphs on n vertices, and F_{n,k}^{∗} the set of pairs (Γ, O), where Γ ∈ F_{n,k} and O is an orientation on Γ —that is, for each vertex v of Γ, O prescribes a cyclic ordering of the edges emanating from O. Let F_{n,k}^{!} denote the subset of F_{n,k} consisting of graphs without loops, double edges, or 3-cycles. As a probability space, F_{n,k}^{!} has positive measure in F_{n,k} bounded away from 0 as n → ∞.

We describe a way of associating to the pair (Γ, O) ∈ F_{n,3} a compact Riemann surface S^{C}(Γ, O). By a theorem of Belyi [2], the surfaces that arise in this way are dense in the space of all compact Riemann surfaces.

We then have:

Theorem 2 ([11]). There exists a constant C_{1} with the following property:

(a) If Γ is picked randomly from F_{n,4}^{!} for n odd, then with probability → 1 as n → ∞, there is an orientation O on Γ so that for any splitting of (Γ, O) to a 3-regular graph (Γ′, O′), the surface S^{C}(Γ′, O′) satisfies:

(i) S^{C}(Γ′, O′) has genus \frac{n+1}{2}; (ii) λ_{1}(S^{C}(Γ′, O′)) ≥ C_{1}.

(b) If Γ is picked randomly from F_{n,3}^{!} with n ≡ 2 (mod 4), then with probability → 1 as n → ∞, there is an orientation O on Γ such that the surface S^{C}(Γ′, O′) satisfies:

(i) S^{C}(Γ′, O′) has genus \frac{n+2}{4}; (ii) λ_{1}(S^{C}(Γ′, O′)) ≥ C_{1}.

Theorem 3 ([12]). There exist constants C_{2}, C_{3}, C_{4}, and C_{5} with the following property: if (Γ, O) is picked randomly from F_{n,3}^{∗}, then, as n → ∞, the surface S^{C}(Γ, O) will have the following properties, with probability at least C_{2}:

(a) λ_{1}(S^{C}(Γ, O)) ≥ C_{3}.

(b) The length of the shortest geodesic syst(S^{C}(Γ, O)) of S^{C}(Γ, O) satisfies

syst(S^{C}(Γ, O)) ≥ C_{4}.

(c) The diameter diam(S^{C}(Γ, O)) satisfies

diam(S^{C}(Γ, O)) ≤ C_{5} log(genus(S^{C}(Γ, O))).

Acknowledgements

The first author would like to thank the UMPA of the École Normale Supérieure of Lyon, for its hospitality during the time when this paper was written.

1. Open and closed Riemann surfaces

Let S^{O} be an open Riemann surface, carrying a complete metric of constant curvature −1 and finite area. Then there is a unique compactification S^{C} of S^{O} whose conformal structure is uniquely determined from S^{O}. In general, S^{C} need not carry a hyperbolic metric, but under favorable circumstances it will, and indeed the hyperbolic metrics on S^{O} and S^{C} will be closely related.

Theorem 4 ([6]). For all ϵ, there exists L such that, if all the cusps of S^{O} have length ≥ L, then there are canonically defined cusp neighborhoods {U_{i}} on S^{O} and S^{C} such that the hyperbolic metrics ds_{O}^{2} and ds_{C}^{2} satisfy

\frac{1}{(1 + ϵ)}ds_{O}^{2} ≤ ds_{C}^{2} ≤ (1 + ϵ)ds_{O}^{2}

outside the sets {U_{i}}.

See [6] for a precise statement. The length of a cusp is the length of the longest closed horocycle about the cusp.

The inverse process was described in [10]:

Theorem 5 ([10]). Given L, there exisis a number R with the following property: If S is a compact Riemann surface, and {p_{1}, …, p_{k}} points on S such that

(a) The injectivity radius about each point is at least R, and

(b) The balls B(p_{i}, R) of radius R about the points p_{i} are pairwise disjoint, then S − {p_{1}, …, p_{k}} has cusps of length ≥ L.

Using this, one shows:

Theorem 6 ([6, 10]). For L sufficiently large, there is a constant C(L) such that: (i) The Cheeger constants h(S^{O}) and h(S^{C}) satisfy:

\frac{1}{C(L)}h(S^{O}) ≤ h(S^{C}) ≤ C(L)h(S^{O}).

(ii) The first eigenvalues λ_{1}(S^{O}) and λ_{1}(S^{C}) satisfy

\frac{1}{C(L)}λ_{1}(S^{O}) ≤ λ_{1}(S^{C}) ≤ C(L)λ_{1}(S^{O}).

(iii) The shortest closed geodesics satisfy

\frac{1}{C(L)} syst(S^{O}) ≤ syst(S^{C}) ≤ C(L) syst(S^{O}).

Furthermore, C(L) → 1 as L → ∞.

By combining this result with the technique of [15] of closing off cusps by forming handles, we prove

Theorem 7 ([10]). Let {S_{i}} be a collection of compact Riemann surfaces with the following properties:

(a) There exists λ > 0 such that λ_{1}(S_{i}) > λ for all i.

(b) syst(S_{i}) → ∞ as i → ∞.

(c) For every C > 1 there exists an N such that

{x ∈ ℝ : x > N} ⊂ ⋃_{i} [1, C](genus(S_{i})).

Then, for all ϵ, there exists N such that for g ≥ N there exists a surface S_{g} of genus g with

λ_{1}(S_{g}) ≥ λ − ϵ.

Applying this theorem to the compactifications of the modular surfaces then gives Theorem 1.

Figure 1. The marked ideal triangle T.

2. Riemann surfaces and 3-regular graphs

Let Γ be a 3-regular graph. An orientation O of Γ is an assignment to each vertex v of Γ of a cyclic ordering of the edges emanating from v. It is clear that a 3-regular graph on n vertices possesses 2^{n} orientations.

To the pair (Γ, O) we will assign two surfaces S^{O}(Γ, O) and S^{C}(Γ, O). The surface S^{O}(Γ, O) is obtained by gluing one copy of the ideal hyperbolic triangle T shown in Figure 1 for each vertex of Γ, such that the natural orientation of the geodesic segments on T matches up with the orientation about the vertex. Whenever two vertices are joined by an edge, we glue the corresponding triangles together, subject to the following conditions:

(i) The geodesic segments on the copies of T are glued together. (ii) The orientations on the copies of T (as complex manifolds with boundary)

are preserved.

The surfaces S^{C}(Γ, O) are the compactifications of the surfaces S^{O}(Γ, O).

As discussed in [8], the geometry and even the topology of the surfaces S^{O}(Γ, O) and S^{C}(Γ, O) depend very strongly on the orientation O. We will say that a path γ on (Γ, O) is a left-hand-turn path if, whenever it arrives at a vertex, it turns left according to the orientation O. Then each cusp of S^{O}(Γ, O) is associated to a unique left-hand-turn path. If we denote by #(LHT) the number of these paths, then the genus of S^{C}(Γ, O) is clearly

genus(S^{C}(Γ, O)) = 1 + \frac{n − 2#(LHT)}{4}.

The length of the cusp corresponding to a given left-hand-turn path γ is precisely the number of edges in γ.

Many geometric properties of the surface S^{O}(Γ, O) are reflected in the pair (Γ, O). Some of these properties follow from general properties of covering manifolds ([7] and [9]). Other properties depend more delicately on this particular construction. When the graph has no short left-hand-turn paths, then these properties descend to properties on S^{C}(Γ, O) via Theorem 6.

Theorem 8. For some L sufficiently large, there are constants C_{1}, …, C_{6} with the following property: Suppose that (Γ, O) has no left-hand-turn paths of length ≤ L. Then

(i) C_{1}λ_{1}(Γ) ≤ λ_{1}(S^{C}(Γ, O)) ≤ C_{2}λ_{1}(Γ). (ii) C_{3}h(Γ) ≤ S^{C}(Γ, O) ≤ C_{4}h(Γ). (iii) log(syst(Γ)) ≤ syst(S^{C}(Γ, O)) ≤ C_{5} syst(Γ). (iv) ([11]) diam(S^{C}(Γ, O)) ≤ C_{6} diam(Γ).

With such strong control over the surface S^{C}(Γ, O), one might be led to expect that the surfaces S^{C}(Γ, O) are rather rare. It is therefore rather surprising that they in fact are quite common.

Theorem 9 ([2, 12]). Given any compact Riemann surface S, there are arbitrarily small deformations S_{ϵ} of S such that S_{ϵ} = S^{C}(Γ, O) for some pair (Γ, O).

3. Models of random graphs

Theorems 2 and 3 are now obtained by an analysis of the process of picking a random graph. To carry out this analysis, we make use of the model of random graphs considered by Bollobás [3, 4]. In this model, a k-regular graph on n vertices is constructed at random by putting nk balls into a hat, k balls for each vertex. The balls are drawn out of the hat in pairs, and an edge drawn between v_{1} and v_{2} each time a pair of balls corresponding to v_{1} and v_{2} is drawn. An orientation on the graph may be determined by the order in which the corresponding pairs are drawn.

We will need the following results of [3] and [4]:

Theorem 10. (i) ([4]) There is a constant C_{1} such that, as n → ∞, the proba-

bility that H(Γ) ≥ C_{1} tends to 1. (ii) ([3]) Let X_{1}, …, X_{L} denote the random variable

X_{j} = the number of closed paths of length j in Γ.

Then, for L fixed and n → ∞, the variables X_{1}, …, X_{L} tend to independent Poisson distributions.

To establish Theorem 2 (a), we seek the probability that a randomly chosen 4regular graph will have an orientation with precisely one left-hand-turn path. Using ideas of [20, 21], it is shown in [11] that this will happen with probability → 1 as long as Γ has no closed loops of length 1. The proof of Theorem 2 (b) is similar, using [19] in place of [20, 21].

To establish Theorem 3, we estimate the probability that the pair (Γ, O) has no left-hand-turn paths of length ≤ L. This will certainly be the case if it has no closed paths of length ≤ L whatsoever, from which Theorem 3 follows. By refining this argument, we may get substantially better estimates for the constant C_{2}.

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Department of Mathematics, Technion—Israel Institute of Technology, Haifa, Israel

E-mail address: rbrooks@tx.technion.ac.il

Department of Mathematics and Computer Science, Drake University, Des Moines, IA 50311

Current address: Department of Mathematics, Dartmouth College, Hanover, NH

E-mail address: eranm@math.huji.ac.il