Harmonic maps and the topology of manifolds with positive spectrum and stable minimal hypersurfaces
The paper identifies two more sets of geometric restrictions to a Riemannian manifold \(M^m\) which allow to prove a Liouville type theorem for harmonic maps to complete manifolds \(N^n\) of nonpositive sectional curvature. (1) Suppose \(M\) is a complete non-compact Riemannian manifold of dimension \(\geq2\). Assume that \(\lambda_1(M)>0\) and the Ricci curvature of \(M\) is bounded from below by \(-(1+{1\over2mn})\lambda_1(M)+\delta\). Then any harmonic map \(M\to N\) with finite energy is constant. (2) The same conclusion holds if \(M\) is a complete oriented noncompact stable minimal hypersurface in a complete Riemannian manifold of non-negative bi-Ricci curvature. The ususal topological consequences of such Liouville theorems hold.
scientific article; zbMATH DE number 5140580