For each n 2 3, we present a family of Riemannian metrics g on W” such that each Riemannian manifold M” = (IT’, g) has positive bottom of the spectrum of Laplacian A, (M”) > 0 and bounded geometry 1 K 1 < C but M” admits no non-constant bounded harmonic functions. These Riemannian manifolds mentioned above give a negative answer to a problem addressed by Schoen-Yau [ 181 in dimension n > 3.