On a maximum principle for minimal surfaces and their integrable discrete counterparts

A novel maximum principle for both classical and discrete minimal surfaces is recorded. In the discrete setting, the maximum principle is based on purely geometric notions of discrete Gaußian and mean curvatures and parallel discrete surfaces. As an additional confirmation of the validity of these notions, a discrete analogue of a classical theorem for linear Weingarten surfaces is obtained. Connections with the ‘parallel surface method’ utilized in condensed matter physics are discussed.