Second Robin eigenvalue bounds for Schrödinger operators on Riemannian surfaces

Let $({\mathrm{\Sigma }}^2,d{s}^2)$ be a compact Riemannian surface, possibly with boundary, and consider Schrödinger-type operators of the form $L=\mathrm{\Delta }+V-aK$ together with natural Robin and Steklov-type boundary conditions incorporating a boundary potential $W$ and (in the curvature-corrected setting) the geodesic curvature ${\kappa }_g$ of $\mathrm{\partial }\mathrm{\Sigma }$. Our main contribution is a geometric upper bound for the second Robin eigenvalue in terms of the topology of $\mathrm{\Sigma }$ and the integrals of $V$ and $W$, obtained via a Hersch balancing argument on the capped surface. As a geometric application, we derive sharp topological restrictions for compact two-sided free boundary minimal surfaces of Morse index at most one inside geodesic balls of negatively curved pinched Cartan–Hadamard $3$-manifolds under a mild radius condition. We also prove complementary upper bounds for first eigenvalues in the closed and Robin settings, including rigidity in the curvature-corrected case, and we establish Steklov-type estimates in a coercive regime where the Dirichlet-to-Neumann operator is well defined for all boundary data.