American Institute of Mathematical Sciences

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  2014, 21: 19-27. doi: 10.3934/era.2014.21.19

The spectral gap of graphs and Steklov eigenvalues on surfaces

Bruno Colbois 1, and  Alexandre Girouard 2,

1. 

Université de Neuchâtel, Institut de Mathématiques, Rue Emile-Argand 11, Case postale 158, 2009 Neuchâtel, Switzerland
2. 

Département de mathématiques et de statistique, Université Laval, Pavillon Alexandre- Vachon, 1045, av. de la Médecine, Quebec Qc G1V 0A6, Canada

Received  October 2013 Revised  January 2014 Published  February 2014

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.
Keywords: eigenvalue inequalities, expander graphs., Riemannian surface, Steklov problem.
Mathematics Subject Classification: Primary 58J50, Secondary 35P1.
Citation: Bruno Colbois, Alexandre Girouard. The spectral gap of graphs and Steklov eigenvalues on surfaces. Electronic Research Announcements, 2014, 21: 19-27. doi: 10.3934/era.2014.21.19
References:
[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.  Google Scholar

[2]

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.  Google Scholar

[3]

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

[4]

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.  Google Scholar

[5]

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

[6]

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

[7]

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.  Google Scholar

[8]

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.  Google Scholar

[9]

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

[10]

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.  Google Scholar

[11]

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.  Google Scholar

[12]

M. Karpukhin, Large Steklov and Laplace eigenvalues,, in preparation., ().   Google Scholar

[13]

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

[14]

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.  Google Scholar

[15]

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

[16]

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.  Google Scholar

show all references

References:
[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.  Google Scholar

[2]

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.  Google Scholar

[3]

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

[4]

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.  Google Scholar

[5]

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

[6]

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

[7]

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.  Google Scholar

[8]

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.  Google Scholar

[9]

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

[10]

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.  Google Scholar

[11]

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.  Google Scholar

[12]

M. Karpukhin, Large Steklov and Laplace eigenvalues,, in preparation., ().   Google Scholar

[13]

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

[14]

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.  Google Scholar

[15]

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

[16]

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.  Google Scholar

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