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

Let be a closed embedded minimal hypersurface in the unit sphere S^m+1 and let =_|A| be the norm of its second fundamental form. In this work we prove that the first eigenvalue of the Laplacian of satisfies ₁ () > m2+m (m+1) 32 (12+m+11) ²+8, and ₁ () =m, when. 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 in the spirit of choi1983first.