Upper bounds for the first eigenvalue of the Laplacian on non-orientable surfaces

TL;DR

This paper extends and improves upper bounds for the first eigenvalue of the Laplacian on non-orientable surfaces, building on prior work for orientable surfaces and formalizing earlier approaches.

Contribution

It formalizes Li and Yau's approach for non-orientable surfaces and provides improved bounds for the first Laplacian eigenvalue.

Findings

Formalized Li and Yau's method for non-orientable surfaces

Derived improved upper bounds for the first eigenvalue

Extended the conformal volume approach to non-orientable cases

Abstract

In 1980 Yang and Yau~\cite{YY} proved the celebrated upper bound for the first eigenvalue on an orientable surface of genus $\gamma$. Later Li and Yau~\cite{LY} gave a simple proof of this bound by introducing the concept of conformal volume of a Riemannian manifold. In the same paper they proposed an approach for obtaining a similar estimate for non-orientable surfaces. In the present paper we formalize their argument and improve the bounds stated in~\cite{LY}.