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Second Robin eigenvalue bounds for Schrödinger operators on Riemannian surfaces

Railane Antonia, Marcos P. Cavalcante, Vinicius Souza

TL;DR

This work derives sharp, topology-driven bounds for Robin and Steklov eigenvalues of Schrödinger-type operators on compact Riemannian surfaces, linking spectral data to genus and boundary data via conformal/capping techniques and a Hersch balancing argument. It uses capped-surface conformal mappings to S^2 (or hemispheres) with degree-control to construct test functions, enabling upper bounds for μ1 and σ1 that depend on χ(Σ), γ, r, and integrals of V and W, with rigidity when equality occurs. A novel curvature-aware application shows topological restrictions for two-sided free boundary minimal surfaces with Morse index at most one inside geodesic balls of negatively curved pinched Cartan–Hadamard 3-manifolds under a radius condition, connecting spectral data to curvature pinching and Gauss equations. The paper also develops a Jacobi–Steklov framework in a coercive regime to obtain upper bounds for the first and second Steklov eigenvalues, including sharper planar-domain bounds and curvature-corrected rigidity results, enriching the interplay between spectral theory, conformal geometry, and geometric analysis on surfaces.

Abstract

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.

Second Robin eigenvalue bounds for Schrödinger operators on Riemannian surfaces

TL;DR

This work derives sharp, topology-driven bounds for Robin and Steklov eigenvalues of Schrödinger-type operators on compact Riemannian surfaces, linking spectral data to genus and boundary data via conformal/capping techniques and a Hersch balancing argument. It uses capped-surface conformal mappings to S^2 (or hemispheres) with degree-control to construct test functions, enabling upper bounds for μ1 and σ1 that depend on χ(Σ), γ, r, and integrals of V and W, with rigidity when equality occurs. A novel curvature-aware application shows topological restrictions for two-sided free boundary minimal surfaces with Morse index at most one inside geodesic balls of negatively curved pinched Cartan–Hadamard 3-manifolds under a radius condition, connecting spectral data to curvature pinching and Gauss equations. The paper also develops a Jacobi–Steklov framework in a coercive regime to obtain upper bounds for the first and second Steklov eigenvalues, including sharper planar-domain bounds and curvature-corrected rigidity results, enriching the interplay between spectral theory, conformal geometry, and geometric analysis on surfaces.

Abstract

Let be a compact Riemannian surface, possibly with boundary, and consider Schrödinger-type operators of the form together with natural Robin and Steklov-type boundary conditions incorporating a boundary potential and (in the curvature-corrected setting) the geodesic curvature of . Our main contribution is a geometric upper bound for the second Robin eigenvalue in terms of the topology of and the integrals of and , 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 -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.
Paper Structure (13 sections, 14 theorems, 88 equations)

This paper contains 13 sections, 14 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Closed surfaces: a first-eigenvalue bound
  3. Surfaces with boundary: the Robin problem
  4. The first Robin eigenvalue
  5. The second Robin eigenvalue: proof of Theorem \ref{['thm:intro-mu1']}
  6. The case of planar domains
  7. Application to free boundary minimal surfaces
  8. Curvature consequences under pinching.
  9. A Jacobi--Steklov framework
  10. Coercivity and the Dirichlet-to-Neumann map
  11. Variational characterizations
  12. An upper bound for the first Steklov eigenvalue in the curvature-corrected case
  13. An upper bound for the second Steklov eigenvalue

Key Result

Theorem 1.1

Let $(\Sigma^2,ds^2)$ be a compact oriented surface with boundary, of genus $\gamma$. For eq:Robin-intro with $V\in L^\infty(\Sigma)$ and $W\in L^\infty(\partial\Sigma)$,

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 2.1
  • proof
  • Corollary 2.2
  • proof
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • ...and 18 more