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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2601.15213 |
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Table of Contents:
- Let $(Σ^2,ds^2)$ be a compact Riemannian surface, possibly with boundary, and consider Schrödinger-type operators of the form $L=Δ+V-aK$ together with natural Robin and Steklov-type boundary conditions incorporating a boundary potential $W$ and (in the curvature-corrected setting) the geodesic curvature $κ_g$ of $\partialΣ$. Our main contribution is a geometric upper bound for the second Robin eigenvalue in terms of the topology of $Σ$ 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.