Liouville theorem for immersed minimal surfaces in any codimension

For a proper immersed minimal disk in ${\mathbf{R}}^{\mathbf{N}}$ with quadratic area growth, we show that any harmonic function whose negative part grows at a slow sub-linear rate is constant. This leads to a higher codimensional Bernstein theorem for minimal disks contained in a sub-linearly growing cone. The catenoid, helicoid and Enneper’s family of surfaces together show that this result is optimal. We also show uniform Hölder regularity of harmonic functions.