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The first eigenvalue of a compact surface can be made arbitrarily small (even for surfaces of fixed genus); see, for example, [1], [2], [3].3] (and references therefrom).

However, as proved by Sarnak and Xue [4], there are arithmetic examples of (constant negative curvature) compact Riemann surfaces of arbitrarily high genus with the first eigenvalue bounded away from zero (see also [5] for a construction involving Selberg's $3/16$'' theorem).

[1] B. Randol, Small eigenvalues of Laplace operator on compact Riemann surfaces, Bull. AMS 80, 1974 996-1008

[2] R. Schoen, S. Wolpert, S. Yau, Geometric bounds on the low eigenvalues of a compact surface, Proc. Symp. Pure Math, vol. 36, AMS, 1980, 279-285.

[3] P. Buser, On Cheeger’s inequality $λ_1 ≥ \frac{h^2}{4}$. Proc. Symp. Pure. Math. vol. 36, 29–77.

[4] P. Sarnak and X. Xue, Bounds for multiplicites of automorphic representations, Duke Math J. 64, 1991, 207-227.

[5] R. Brooks, E. Makover, Riemann surfaces with large first eigenvalue. J. Anal. Math. 83, 2001, 243–258.

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The first eigenvalue of a compact surface can be made arbitrariliy arbitrarily small (even for surfaces of fixed genus); see for example [1], [2]. 2], [3].

However, as proved by Sarnak and Xue [4], there are arithmetic examples of (constant negative curvature) compact Riemann surfaces of arbitrarily high genus with the first eigenvalue bounded away from zero [3] (see also [4]).5] for a construction involving Selberg's $3/16$'' theorem).

[1] B. Randol, Small eigenvalues of Laplace operator on compact Riemann surfaces, Bull. AMS 80, 1974 996-1008

[2] R. Schoen, S. Wolpert, S. Yau, Geometric bounds on the low eigenvalues of a compact surface, Proc. Symp. Pure Math, vol. 36, AMS, 1980, 279-285.

[3] P. Buser, On Cheeger’s inequality $λ_1 \frac{h^2}{4}$. Proc. Symp. Pure. Math. vol. 36, 29–77.

[4] P. Sarnak and X. Xue, Bounds for multiplicites of automorphic representations, Duke Math J. 64, 1991, 207-227.

[4] 5] R. Brooks, E. Makover, Riemann surfaces with large first eigenvalue. J. Anal. Math. 83, 2001, 243–258.

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The first eigenvalue of a compact Riemann surface can be made arbitrariliy small (even for surfaces of fixed genus); see for example [1], [2]. However there are arithmetic examples of compact Riemann surfaces of arbitrarily high genus with the first eigenvalue bounded away from zero [3]; 3] (see also [4].4]).

[1] B. Randol, Small eigenvalues of Laplace operator on compact Riemann surfaces, Bull. AMS 80, 1974 996-1008

[2] R. Schoen, S. Wolpert, S. Yau, Geometric bounds on the low eigenvalues of a compact surface, Proc. Symp. Pure Math, vol. 36, AMS, 1980, 279-285.

[3] P. Sarnak and X. Xue, Bounds for multiplicites of automorphic representations, Duke Math J. 64, 1991, 207-227.

[4] R. Brooks, E. Makover, Riemann surfaces with large first eigenvalue. J. Anal. Math. 83, 2001, 243–258.

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