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On the first eigenvalue of the Laplacian on compact surfaces of genus three. (English) Zbl 1496.35268

Summary: For any compact Riemannian surface of genus three \((\Sigma, ds^2)\) Yang and Yau proved that the product of the first eigenvalue of the Laplacian \(\lambda_1(ds^2)\) and the area \(\mathit{Area}(ds^2)\) is bounded above by \(24\pi\). In this paper we improve the result and we show that \(\lambda_1(ds^2) \mathit{Area}(ds^2) \leq 16(4 - \sqrt{7})\pi \approx 21.668\pi\). About the sharpness of the bound, for the hyperbolic Klein quartic surface numerical computations give the value \(\approx 21.414\pi\).

MSC:

35P15 Estimates of eigenvalues in context of PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
58C40 Spectral theory; eigenvalue problems on manifolds
58J05 Elliptic equations on manifolds, general theory
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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References:

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