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(12Λ+m+11)²+8, and λ₁(Σ)=m, when Λ≤√m. In particular, this estimate improves the one obtained recently in {duncan2023improved}. 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 {choi1983first}.