For any compact Riemannian surface of genus three (E, ds2) Yang and Yau proved that the product of the first eigenvalue of the Laplacian Ai(ds2) and the area Area(ds2) is bounded aboye by 247r. In this paper we improve the result and we show that Ai(ds2)Area(ds2) < 16(4 ??? Nr7)7r 21.6687r. About the sharpness of the bound, for the hyperbolic Klein quartic surface numerical computations give the value 21.4147r.
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