The first eigenvalue of the Laplacian on orientable surfaces

TL;DR

This paper generalizes methods to improve bounds on the first Laplacian eigenvalue on orientable surfaces using holomorphic maps to complex projective spaces, refining previous inequalities for many genera.

Contribution

It extends Ros's approach by employing holomorphic maps to cf^n for all n>0, leading to improved inequalities for a broad range of surface genera.

Findings

Improved bounds for eigenvalues for most genera .

Generalization to holomorphic maps to cf^n.

Excludes specific genera where bounds are not improved.

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 $\gamma$ and the area. Its proof relies on the existence of holomorhic maps to $\mathbb{CP}^1$ of low degree. Very recently, A.~Ros was able to use certain holomorphic maps to $\mathbb{CP}^2$ in order to give a quantitative improvement of the Yang-Yau inequality for $\gamma=3$. In the present paper, we generalize Ros' argument to make use of holomorphic maps to $\mathbb{CP}^n$ for any $n>0$. As an application, we obtain a quantitative improvement of the Yang-Yau inequality for all genera $\gamma>3$ except for $\gamma = 4,6,8,10,14$.