For a bounded Lipschitz domain \(\Sigma \) in a Riemannian surface M satisfying certain curvature condition, we prove that \(\begin{aligned} \mu _{3-\beta _1} \le \lambda _{1}, \end{aligned}\) where \(\mu _k\) ( \(\lambda _k\) resp.) is the k-th Neumann (Dirichlet resp.) Laplacian eigenvalue on \(\Sigma \) and \(\beta _1\) is the first Betti number of \(\Sigma .\) If \(\Sigma \) is smooth and simply connected, we can further derive the strict inequality \( \mu _{3}< \lambda _{1}. \) This extends previous results on the Euclidean space to various curved surfaces, including the flat cylinder, the hyperbolic plane, hyperbolic cusp, collar, funnel, and minimal surfaces such as catenoid and helicoid. The novelty of the paper lies in comparing Dirichlet and Neumann Laplacian eigenvalues via the variational principle of the Hodge Laplacian on 1-forms on a surface, extending the variational principle on vector fields in the Euclidean plane as developed by Rohleder (Math Ann 392(4):5553–5571, 2025). The comparison is reduced to the existence of a distance function with appropriate curvature conditions on its level sets.
