Upper Bounds for the First Eigenvalue of the Laplacian on Non-Orientable Surfaces

In 1980, Yang and Yau [20] proved the celebrated upper bound for the first eigenvalue on an orientable surface of genus $\gamma $. Later Li and Yau [14] 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 [14].