Gamow liquid-drop minimizer conjecture

For a measurable set $\Omega\subset\mathbb R^3$, let $$\mathcal E(\Omega)=P(\Omega)+\frac12\iint_{\Omega\times\Omega}\frac{dx\,dy}{|x-y|},$$ where $P$ is De Giorgi perimeter, and set $$V_*=5\frac{2-2^{2/3}}{2^{2/3}-1}\approx3.51.$$ The conjecture asks for the complete fixed-volume minimization picture. Chodosh and Gianocca prove that, for every $0<V\le V_*$, balls of volume $V$ uniquely minimize $\mathcal E$ among all measurable $\Omega$ with $|\Omega|=V$, up to translation and null sets; for $V>V_*$, no minimizer exists. Consequently, $$\inf_{0<|\Omega|<\infty}\frac{\mathcal E(\Omega)}{|\Omega|}=3\left(\frac{9\pi}{5}\right)^{1/3}=\frac92\left(\frac{8\pi}{15}\right)^{1/3},$$ with equality exactly for translates, modulo null sets, of the ball of volume $5/2$, equivalently radius $(15/(8\pi))^{1/3}$.

The complete fixed-volume picture, closing a gap that partial results had narrowed from both ends without meeting: balls uniquely minimize for every volume up to V_* = 3.51..., and above it no minimizer exists at all. Before this the best minimality range was V <= 1 (Chodosh-Ruohoniemi, 2025) and the best nonexistence bound V >= 7.5 (Schulz, posted two days earlier), so the open middle ran from 1 to 7.5. Frank-Nam had already proved existence up to V_*, and the new proof uses it; the fresh content is uniqueness of the ball across the whole range and nonexistence immediately above the threshold. A corollary settles the minimal binding energy question of Frank-Lieb: the infimum of E(Omega)/|Omega| is 3(9pi/5)^(1/3), attained exactly at balls of volume 5/2. The mechanism is a capacitary estimate that sharpens an Agostiniani-Mazzieri monotonicity formula using Gauss-Bonnet, an improvement the authors note applies only to this particular weight and only in three dimensions.