The first eigenvalue of the Laplacian on orientable surfaces

The famous Yang–Yau inequality provides an upper bound for the first eigenvalue of the Laplacian on an orientable Riemannian surface solely in terms of its genus γ and the area. Its proof relies on the existence of holomorhic maps to ℂℙ^1 of low degree. Very recently, Ros was able to use certain holomorphic maps to ℂℙ^2 in order to give a quantitative improvement of the Yang–Yau inequality for γ =3 . In the present paper, we generalize Ros’ argument to make use of holomorphic maps to ℂℙ^n for any n>0 . As an application, we obtain a quantitative improvement of the Yang–Yau inequality for all genera except for γ = 4,6,8,10,14 .