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The spectral gap of graphs and Steklov eigenvalues on surfaces

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  • Using expander graphs, we construct a sequence $\{\Omega_N\}_{N\in\mathbb{N}}$ of smooth compact surfaces with boundary of perimeter $N$, and with the first non-zero Steklov eigenvalue $\sigma_1(\Omega_N)$ uniformly bounded away from zero. This answers a question which was raised in [10]. The sequence $\sigma_1(\Omega_N) L(\partial\Omega_n)$ grows linearly with the genus of $\Omega_N$, which is the optimal growth rate.
    Mathematics Subject Classification: Primary 58J50, Secondary 35P15.


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  • [1]

    R. Bañuelos, T. Kulczycki, I. Polterovich and B. Siudeja, Eigenvalue inequalities for mixed Steklov problems, in Operator Theory and its Applications, Amer. Math. Soc. Transl. Ser. 2, 231, Amer. Math. Soc., Providence, RI, 2010, 19-34.


    R. Brooks, The first eigenvalue in a tower of coverings, Bull. Amer. Math. Soc. (N.S.), 13 (1985), 137-140.doi: 10.1090/S0273-0979-1985-15397-2.


    R. Brooks, The spectral geometry of a tower of coverings, J. Differential Geom., 23 (1986), 97-107.


    M. Burger, Estimation de petites valeurs propres du laplacien d'un revêtement de variétés riemanniennes compactes, C. R. Acad. Sci. Paris Sér. I Math., 302 (1986), 191-194.


    P. Buser, On the bipartition of graphs, Discrete Appl. Math., 9 (1984), 105-109.doi: 10.1016/0166-218X(84)90093-3.


    F. R. K. Chung, Spectral Graph Theory, CBMS Regional Conference Series in Mathematics, No. 92, American Mathematical Society, Providence, RI, 1997.


    B. Colbois, A. El Soufi and A. Girouard, Isoperimetric control of the Steklov spectrum, J. Funct. Anal., 261 (2011), 1384-1399.doi: 10.1016/j.jfa.2011.05.006.


    B. Colbois and A.-M. Matei, On the optimality of J. Cheeger and P. Buser inequalities, Differential Geom. Appl., 19 (2003), 281-293.doi: 10.1016/S0926-2245(03)00035-4.


    A. Fraser and R. Schoen, Sharp eigenvalue bounds and minimal surfaces in the ball, arXiv:1209.3789, (2013).


    A. Girouard and I. Polterovich, Upper bounds for Steklov eigenvalues on surfaces, Electron. Res. Announc. Math. Sci., 19 (2012), 77-85.doi: 10.3934/era.2012.19.77.


    S. Hoory, N. Linial and A. Wigderson, Expander graphs and their applications, Bull. Amer. Math. Soc. (N.S.), 43 (2006), 439-561 (electronic).doi: 10.1090/S0273-0979-06-01126-8.


    M. Karpukhin, Large Steklov and Laplace eigenvalues, in preparation.


    G. Kokarev, Variational aspects of Laplace eigenvalues on Riemannian surfaces, arXiv:1103.2448, (2011).


    N. N. Moiseev, Introduction to the theory of oscillations of liquid-containing bodies, in Advances in Applied Mechanics, Vol. 8, Academic Press, New York, 1964, 233-289.


    M. S. Pinsker, On the complexity of a concentrator, in 7th International Teletraffic Conference, 1973, 318/1-318/4.


    P. C. Yang and S. T. Yau, Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 7 (1980), 55-63.

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