On the Shape of Liquid Drops and Crystals in the Small Mass Regime

We consider liquid drops or crystals lying in equilibrium under the action of a potential energy. For small masses, the proximity of the resulting minimizers from the Wulff shape associated to the surface tension is quantitatively controlled in terms of the smallness of the mass and with respect to the natural notions of distance induced by the regularity of the Wulff shape. Stronger results are proved in the two-dimensional case. For instance, it is shown that a planar crystal undergoing the action of a small exterior force field remains convex, and admits only small translations parallel to its faces.