On the three-dimensional shape of a crystal

In this paper we completely settle the Almgren problem in $\mathbb R^3$ under some generic conditions on the potential and tension functions. The problem, among other things, appears in classical thermodynamics when one is to understand if minimizing the free energy with convex potential and under a mass constraint generates a convex crystal. Our new idea in proving a three-dimensional convexity theorem is to utilize a stability theorem when $m$ is small, convexity when $m$ is small, and the first variation PDE with a new maximum principle approach.