arXiv:math/0608342v1 [math.DG] 14 Aug 2006 A remark on compact CMC-Hypersurfaces of N × R G. Pacelli Bessa J. Fabio Montenegro August 26, 2018 Abstract Let F be the set of all closed hypersurfaces M immersed in N × R with constant mean curvature H M , where N is a simply connected complete Riemannian n-manifold with sectional curvature K N ≤ κ< 0. We show that inf M∈F |H M |≥ (n − 1)κ/n. That answers a question posed by H. Rosenberg. Mathematics Subject Classification: (2000): 53C40, 53C42, 58C40 Key words: constant mean curvature, fundamental tone, extrinsic radius. 1 Introduction Let M = {(x, f (x))}⊂ N × R be a graph of a smooth function f : N → R with constant mean curvature H M then it follows from the work of Barbosa, Kenmotsu and Oshikiri in [1] and Salavessa in [14] that n ·|H M |≤ 2 ·  λ ∗ (N ). (1) In particular, if N is a simply connected complete Riemannian manifold with sectional curvature K N ≤−κ 2 < 0, κ> 0, then by McKean’s Theorem and by (1) we have that n ·|H M |≤ 2 ·  λ ∗ (H n (−κ 2 )) = (n − 1) · κ (2) Nelli and Rosenberg in [11] for n=2 and Salavessa in [13], [14] for any n, constructed entire graphs M = {(x, f (x))}⊂ H n (−1) × R with constant mean curvature |H M | = c/n, for each c ∈ (0,n − 1]. Thus, by the maximum principle any closed constant mean curvature hypersurface, (CMC-hypersurface), M ⊂ H n (−1) × R has mean curvature |H M | > (n − 1)/n. It should be noticed that Hsiang and Hsiang [8] also showed that |H M | > 1/2 for any closed CMC-surface M ⊂ H 2 (−1) × R. Let F (N ) be the set of all closed CMC-hypersurfaces M immersed in N × R with mean curvature H M , where N is a complete n-dimensional Riemannian manifold. Harold Rosenberg in [12] among other questions, asked how small can be the mean curvature |H M |, M ∈F (N ). In this paper we answer this question when N is simply connected complete Riemannian manifold with sectional curvature bounded above K N ≤−κ 2 < 0 without constructing any entire graphs with constant mean curvature. We prove the following theorem. 1