The first eigenvalue of the Laplacian on orientable surfaces

Abstract

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 CP¹ of low degree. Very recently, A.~Ros was able to use certain holomorphic maps to CP² 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 γ > 3 except for γ = 4,6,8,10,14.