Second Robin eigenvalue bounds for Schr\"odinger operators on Riemannian surfaces

TL;DR

This paper establishes geometric upper bounds for the second Robin eigenvalue of Schr"odinger operators on Riemannian surfaces, linking spectral properties to topology, boundary conditions, and curvature, with applications to minimal surfaces in negatively curved manifolds.

Contribution

It provides new geometric bounds for Robin eigenvalues of Schr"odinger operators on surfaces, incorporating boundary conditions and curvature, and applies these results to minimal surface theory in curved 3-manifolds.

Findings

Upper bounds for second Robin eigenvalues based on topology and integrals of potentials.

Sharp topological restrictions for minimal surfaces with low Morse index.

Rigidity results and Steklov-type estimates in curvature-corrected settings.

Abstract

Let $(\Sigma^2,ds^2)$ be a compact Riemannian surface, possibly with boundary, and consider Schr\"odinger-type operators of the form $L=\Delta+V-aK$ 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 $\partial\Sigma$. Our main contribution is a geometric upper bound for the second Robin eigenvalue in terms of the topology of $\Sigma$ 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.