  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.
