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 \emph{Operator Theory and its Applications}, (2010), 19.   Google Scholar

[2]

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

[3]

R. Brooks, The spectral geometry of a tower of coverings,, \emph{J. Differential Geom.}, 23 (1986), 97.   Google Scholar

[4]

M. Burger, Estimation de petites valeurs propres du laplacien d'un revêtement de variétés riemanniennes compactes,, \emph{C. R. Acad. Sci. Paris Sér. I Math.}, 302 (1986), 191.   Google Scholar

[5]

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

[6]

F. R. K. Chung, Spectral Graph Theory,, CBMS Regional Conference Series in Mathematics, (1997).   Google Scholar

[7]

B. Colbois, A. El Soufi and A. Girouard, Isoperimetric control of the Steklov spectrum,, \emph{J. Funct. Anal.}, 261 (2011), 1384.  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,, \emph{Differential Geom. Appl.}, 19 (2003), 281.  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,, \emph{Electron. Res. Announc. Math. Sci.}, 19 (2012), 77.  doi: 10.3934/era.2012.19.77.  Google Scholar

[11]

S. Hoory, N. Linial and A. Wigderson, Expander graphs and their applications,, \emph{Bull. Amer. Math. Soc. (N.S.)}, 43 (2006), 439.  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 \emph{Advances in Applied Mechanics, (1964), 233.   Google Scholar

[15]

M. S. Pinsker, On the complexity of a concentrator,, in \emph{7th International Teletraffic Conference}, (1973), 1.   Google Scholar

[16]

P. C. Yang and S. T. Yau, Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)}, 7 (1980), 55.   Google Scholar

show all references

References:
[1]

R. Bañuelos, T. Kulczycki, I. Polterovich and B. Siudeja, Eigenvalue inequalities for mixed Steklov problems,, in \emph{Operator Theory and its Applications}, (2010), 19.   Google Scholar

[2]

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

[3]

R. Brooks, The spectral geometry of a tower of coverings,, \emph{J. Differential Geom.}, 23 (1986), 97.   Google Scholar

[4]

M. Burger, Estimation de petites valeurs propres du laplacien d'un revêtement de variétés riemanniennes compactes,, \emph{C. R. Acad. Sci. Paris Sér. I Math.}, 302 (1986), 191.   Google Scholar

[5]

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

[6]

F. R. K. Chung, Spectral Graph Theory,, CBMS Regional Conference Series in Mathematics, (1997).   Google Scholar

[7]

B. Colbois, A. El Soufi and A. Girouard, Isoperimetric control of the Steklov spectrum,, \emph{J. Funct. Anal.}, 261 (2011), 1384.  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,, \emph{Differential Geom. Appl.}, 19 (2003), 281.  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,, \emph{Electron. Res. Announc. Math. Sci.}, 19 (2012), 77.  doi: 10.3934/era.2012.19.77.  Google Scholar

[11]

S. Hoory, N. Linial and A. Wigderson, Expander graphs and their applications,, \emph{Bull. Amer. Math. Soc. (N.S.)}, 43 (2006), 439.  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 \emph{Advances in Applied Mechanics, (1964), 233.   Google Scholar

[15]

M. S. Pinsker, On the complexity of a concentrator,, in \emph{7th International Teletraffic Conference}, (1973), 1.   Google Scholar

[16]

P. C. Yang and S. T. Yau, Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)}, 7 (1980), 55.   Google Scholar

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