Abstract
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 $Area(ds^2)$ is bounded above by $24\pi$. In this paper we improve the result and we show that $\lambda_1(ds^2) Area(ds^2) \leq 16(4 - \sqrt{7})\hspace{.02cm}\pi \approx 21.668\hspace{.02cm}\pi$. About the sharpness of the bound, for the hyperbolic Klein quartic surface numerical computations give the value $\approx 21.414\hspace{.02cm}\pi$.
Funding Statement
Partially supported by MINECO/FEDER grants no. MTM2017-89677-P and Regional J. Andalucía grants no. P06-FQM-01642 and P18-FR-4049.
Citation
Antonio ROS. "On the first eigenvalue of the Laplacian on compact surfaces of genus three." J. Math. Soc. Japan 74 (3) 813 - 828, July, 2022. https://doi.org/10.2969/jmsj/85898589
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