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arXiv2026

Topology of stable free boundary CMC surfaces under lower Ricci curvature bounds

We establish intrinsic area--length--topology inequalities for compact free boundary constant mean curvature (CMC) surfaces in three-manifolds with Ricci curvature bounded from below. Our main result is obtained from a conformal upper bound for a constrained first Robin eigenvalue of the Jacobi operator, derived via a balancing argument. This yields a quantitative inequality that does not require stability and captures both interior and boundary contributions. As an application, we obtain explicit topological restrictions for stable free boundary CMC surfaces under a natural curvature pinching condition. In particular, in weakly convex domains, stability forces low topological complexity, with genus at most three and a small number of boundary components. These results show that effective topological control persists even in negatively curved settings, where classical rigidity phenomena are no longer available.

math.DG↗

arXiv2026

Second Robin eigenvalue bounds for Schrödinger operators on Riemannian surfaces

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.

math.DG↗