Perturbed cone theorems for proper harmonic maps

Inspired by the halfspace theorem for minimal surfaces in ${\mathbb{R}}^3$ of Hoffman-Meeks, the halfspace theorem of Rodriguez-Rosenberg, and the cone theorem of Omori, we derive new non-existence results for proper harmonic maps into perturbed cones in ${\mathbb{R}}^n$, horospheres in ${\mathbb{H}}^n$ and also into perturbed Riemannian cones. The technical tool in use is an extension of the foliated maximum principle appearing in Assimos-Jost to the non-compact setting.