We investigate the nanometer-scale flow of molten polyethylene from a heated atomic force microscope (AFM) cantilever tip during thermal dip-pen nanolithography (tDPN). Polymer nanostructures were written for cantilever tip temperatures and substrate temperatures controlled over the range 100-260 °C and while the tip was either moving with speed 0.5-2.0 µm s(-1) or stationary and heated for 0.1-100 s. We find that polymer flow depends on surface capillary forces and not on shear between tip and substrate. The polymer mass flow rate is sensitive to the temperature-dependent polymer viscosity. The polymer flow is governed by thermal Marangoni forces and non-equilibrium wetting dynamics caused by a solidification front within the feature.