Second Robin eigenvalue bounds for Schrödinger operators on Riemannian surfaces
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| Format: | Preprint |
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2026
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| _version_ | 1866912851278430208 |
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| author | Antonia, Railane Cavalcante, Marcos P. Souza, Vinicius |
| author_facet | Antonia, Railane Cavalcante, Marcos P. Souza, Vinicius |
| 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_15213 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Second Robin eigenvalue bounds for Schrödinger operators on Riemannian surfaces Antonia, Railane Cavalcante, Marcos P. Souza, Vinicius Differential Geometry Primary 35P15, 35J10, 58J50, Secondary 53A10, 49Q05 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. |
| title | Second Robin eigenvalue bounds for Schrödinger operators on Riemannian surfaces |
| topic | Differential Geometry Primary 35P15, 35J10, 58J50, Secondary 53A10, 49Q05 |
| url | https://arxiv.org/abs/2601.15213 |