Complete hypersurfaces in a Euclidean space Rn+1Rn+1 with constant mth mean curvature

In this paper, we consider n-dimensional oriented complete hypersurfaces with constant m  th mean curvature of a Euclidean space Rn+1Rn+1. We characterize the hypersurface Sk(c)×Rn−kSk(c)×Rn−k in a Euclidean space Rn+1Rn+1 and show that generalized Yau conjecture is true for the class of oriented compact locally conformally flat hypersurfaces with positive constant m  th mean curvature of a Euclidean space Rn+1Rn+1. When m=2m=2, our results reduce to the results of Q.-M. Cheng [Q.M. Cheng, Complete hypersurfaces in a Euclidean space Rn+1Rn+1 with constant scalar curvature, Indiana Univ. Math. J. 51 (2002) 53–68].