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By introducing a more flexible notion of convexity, we obtain a new Omori-Yau maximum principle for harmonic maps. In the spirit of the Calabi-Yau conjectures, this principle is more suitable for studying the unboundedness of certain totally geodesic projections of minimal submanifolds of higher codimension. We further explore this maximum principle by applying it to conformal maps, harmonic maps into Cartan-Hadamard manifolds, as well as cone, wedge and halfspace theorems., Comment: 28 pages, 6 figures

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., Comment: 19 pages, 7 figures