An Improved Eigenvalue Estimate for Embedded Minimal Hypersurfaces in the Sphere

Suppose that Σ ⁿ⊂ Sⁿ⁺¹ is a closed embedded minimal hypersurface. We prove that the first non-zero eigenvalue λ ₁ of the induced Laplace–Beltrami operator on Σ satisfies λ ₁ ≥ n/2+ aₙ(Λ ⁶ + bₙ)⁻¹, where aₙ and bₙ are explicit dimensional constants and Λ is an upper bound for the length of the second fundamental form of Σ. This provides the first explicitly computable improvement on Choi and Wang’s lower bound λ ₁ ≥ n/2 without any further assumptions on Σ.