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edited May 6 2011 at 10:45 |
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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edited May 6 2011 at 9:27 |
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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edited May 5 2011 at 17:35 |
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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edited May 5 2011 at 17:22 |
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 constant negative curvature and arbitrary arbitrarily high volume genus with first eigenvalue bounded away from zero [3]; see also [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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edited May 5 2011 at 17:13 |
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 constant negative curvature and arbitrary high volume with first eigenvalue bounded away from zero , [3]; see also [3].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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answered May 5 2011 at 16:44 |
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 constant negative curvature and arbitrary high volume with first eigenvalue bounded away from zero, see [3].
[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.
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