A lower bound for the first eigenvalue of a minimal hypersurface in the sphere

Let Σ be a closed embedded minimal hypersurface in the unit sphere Sᵐ⁺¹ and let Λ=maxΣ|A| be the norm of its second fundamental form. In this work, we prove that the first eigenvalue of the Laplacian of Σ satisfies λ₁(Σ) > m/2+{m(m+1)}{32 (12 Λ+m+11)²+8}, and λ₁(Σ)=m when Λ≤√m . In particular, this estimate improves the one obtained recently in Duncan–Sire–Spruck (2024). The proof of our main result is based on a Rayleigh quotient estimate for a harmonic extension of an eigenfunction of the Laplacian of Σ in the spirit of Choi and Wang (1983).