Examples of simply-connected Liouville manifolds with positive spectrum
Let \(n\geq 3\). The authors present a family of Riemannian metrics \(g\) on Euclidean space \(\mathbb{R}^n\) so that \((\mathbb{R}^n,g)\) has positive bottom of the spectrum of the Laplacian, i.e., \(\lambda_1(\mathbb{R}^n,g)>0\), and bounded geometry \(|K|\leq C\), but \((\mathbb{R}^n,g)\) has no nonconstant bounded harmonic functions. This gives a negative answer to a problem posed by \textit{R. Schoen} and \textit{S.-T. Yau} [Lectures on differential geometry. Cambridge, MA: International Press (1994; Zbl 0830.53001)]. If \(n=3\), these metrics are double warped product metrics of the form \(g=e^{2\eta_1(s)}dx^2+e^{2\eta_2(s)}dy^2+ds^2\) where the \(\eta_i\) are suitably chosen.
scientific article; zbMATH DE number 897464
scientific article