The Fundamental Gap of Simplices
The Fundamental Gap of Simplices
复制标题
单纯形的基本差距
DOI：
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
J. Rowlett
中科院分区：
文献类型：
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作者：
Zhiqin Lu;J. Rowlett
AbstractThe gap function of a domain $${Omega subset mathbb{R}^n}$$ is$$xi(Omega) := d^2 (lambda_2 - lambda_1)$$, where d is the diameter of Ω, and λ1 and λ2 are the first two positive Dirichlet eigenvalues of the Euclidean Laplacian on Ω. It was recently shown by Andrews and Clutterbuck (J Amer Math Soc 24:899–916, 2011) that for any convex $${Omega subset mathbb{R}^n}$$,$$xi(Omega) geq 3 pi^2$$, where the infimum occurs for n = 1. On the other hand, the gap function on the moduli space of n-simplices behaves differently. Our first theorem is a compactness result for the gap function on the moduli space of n-simplices. Next, specializing to n = 2, our second main result proves the recent conjecture of Antunes-Freitas (J Phys A: Math Theor 41(5):055201, 2008) for any triangle$${T subset mathbb{R}^2}$$,
$$xi(T) geq frac{64 pi^2}{9}$$, with equality if and only if T is equilateral.
AbstractThe gap function of a domain $${Omega subset mathbb{R}^n}$$ is$$xi(Omega) := d^2 (lambda_2 - lambda_1)$$, where d is the diameter of Ω, and λ1 and λ2 are the first two positive Dirichlet eigenvalues of the Euclidean Laplacian on Ω. It was recently shown by Andrews and Clutterbuck (J Amer Math Soc 24:899–916, 2011) that for any convex $${Omega subset mathbb{R}^n}$$,$$xi(Omega) geq 3 pi^2$$, where the infimum occurs for n = 1. On the other hand, the gap function on the moduli space of n-simplices behaves differently. Our first theorem is a compactness result for the gap function on the moduli space of n-simplices. Next, specializing to n = 2, our second main result proves the recent conjecture of Antunes-Freitas (J Phys A: Math Theor 41(5):055201, 2008) for any triangle$${T subset mathbb{R}^2}$$,
$$xi(T) geq frac{64 pi^2}{9}$$, with equality if and only if T is equilateral.