Perturbed cone theorems for proper harmonic maps

Abstract Inspired by the halfspace theorem for minimal surfaces in $\mathbb {R}^3$ of Hoffman–Meeks, the halfspace theorem of Rodriguez–Rosenberg, and the classical cone theorem of Omori in $\mathbb {R}^n$ , we derive new non-existence results for proper harmonic maps into perturbed cones in $\mathbb {R}^n$ , horospheres in $\mathbb {H}^n$ , culminating in a generalization of Omori’s theorem in arbitrary Riemannian manifolds. The technical tool proved here extends the foliated Sampson’s maximum principle, initially developed in the first author’s Ph.D. thesis, to a non-compact setting.