Minimizing the free energy

We prove the sharp quantitative stability in the radial isotropic Almgren problem. In addition, we develop a theory for estimating the sharp modulus in the context of minimal assumptions on the surface tension and the potential and obtain the sharp ${\epsilon }^2$ in any dimension. Inter-alia, we also solve the problem of calculating the critical mass which was only a priori assumed to exist and which breaks the mass regime into two sets: the one where the energy is concave and the one where it is convex.