On the first eigenvalue of the Laplacian on compact surfaces of genus three

Abstract

For any compact Riemannian surface of genus three (Σ,𝑑𝑠²) Yang and Yau proved that the product of the first eigenvalue of the Laplacian 𝜆₁(𝑑𝑠²) and the area 𝐴𝑟𝑒𝑎(𝑑𝑠²) is bounded above by 24𝜋. In this paper we improve the result and we show that 𝜆₁(𝑑𝑠²) 𝐴𝑟𝑒𝑎(𝑑𝑠²) ≤ 16(4 − \sqrt{7})𝜋 ≈ 21.668𝜋. About the sharpness of the bound, for the hyperbolic Klein quartic surface numerical computations give the value ≈ 21.414𝜋.