  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.
