Gamow liquid-drop minimizer conjecture

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.

The central open problem of the liquid-drop literature: the sharp threshold between existence and nonexistence of fixed-volume minimizers, with uniqueness of the ball below it. Not eponymous, and with no Wikipedia article. What lifts it above a single-subfield problem is reach - tracked across calculus of variations, mathematical physics and geometric analysis at once, with a 2017 Notices of the AMS survey for a general audience and a partial-results literature carrying Lieb, Otto, Figalli and Maggi. Hence level with Polya for Neumann balls (35) rather than HRT (33), below the eponymous band.