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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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