Examples of simply-connected Liouville manifolds with positive spectrum

AbstractFor each n ⩾ 3, we present a family of Riemannian metrics g on Rn such that each Riemannian manifold Mn = (Rn, g) has positive bottom of the spectrum of Laplacian λ1(Mn) > 0 and bounded geometry ¦K¦⩽ C but Mn admits no non-constant bounded harmonic functions. These Riemannian manifolds mentioned above give a negative answer to a problem addressed by Schoen-Yau [18] in dimension n ⩾ 3