Eigenvalue estimates for minimal hypersurfaces in hyperbolic space

Recently Candel [A. Candel, Eigenvalue estimates for minimal surfaces in hyperbolic space, Trans. Amer. Math. Soc. 359 (2007) 3567–3575] proved that if M   is a simply-connected stable minimal surface isometrically immersed in H3H3, then the first eigenvalue of M   satisfies 1/4⩽λ(M)⩽4/31/4⩽λ(M)⩽4/3 and he asked whether the bound is sharp and gave an example such that the lower bound is attained. In this note, we prove that the upper bound can never be attained. Also we extend the result by proving that if M   is compact stable minimal hypersurface isometrically immersed in Hn+1Hn+1 where n⩾3n⩾3 such that its smooth Yamabe invariant is negative, then (n−1)/4⩽λ(M)⩽n2(n−2)/(7n−6)(n−1)/4⩽λ(M)⩽n2(n−2)/(7n−6).