On an old conjecture of Almgren

In this talk I will give an overview on the few results available on the conjecture of Almgren regarding the convexity of drops subject to the action of an external convex potential. In particular I will present recent progress in this direction obtained with G. De Philippis on their connectedness. Together with an older result of McCann, this answers positively the conjecture in dimension two. The proof is inspired by the two-point function technique introduced by B. Andrews and is reminiscent of the doubling of variables trick in the context of viscosity solutions.