Let m_operatornameball be the critical mass below which balls minimize the liquid drop energy, let m_ex be the critical mass up to which minimizers exist, and let m_crit be the mass at which the energy of one ball equals twice that of a ball of half the mass. Gamow's conjecture. If n=3 and alpha=1, then m_operatornameball=m_ex=m_crit. Equivalently, the ball of mass m is the unique minimizer up to translations for mleq m_crit, and no minimizer exists for m>m_crit. This conjecture is supported by known results on the liquid drop model and remains open in the Coulomb case; the paper notes that analogous equality is expected in arbitrary dimension and for every 0<alpha<n.

Gamow's conjecture for the Coulomb liquid drop model

Let $m_{\operatorname{ball}}$ be the critical mass below which balls minimize the liquid drop energy, let $m_\mathrm{ex}$ be the critical mass up to which minimizers exist, and let $m_\mathrm{crit}$ be the mass at which the energy of one ball equals twice that of a ball of half the mass. Gamow's conjecture. If $n=3$ and $\alpha=1$ , then

m_{\operatorname{ball}}=m_\mathrm{ex}=m_\mathrm{crit}.

Equivalently, the ball of mass $m$ is the unique minimizer up to translations for $mleq m_{\mathrm{crit}}$ , and no minimizer exists for $m>m_{\mathrm{crit}}$ . This conjecture is supported by known results on the liquid drop model and remains open in the Coulomb case; the paper notes that analogous equality is expected in arbitrary dimension and for every $0<\alpha<n$ .

The conjecture asserts that in three dimensions with Coulomb interaction, the ball is the unique minimizer up to the critical mass and that minimizers do not exist above it, equivalently $m_{\operatorname{ball}}=m_{\mathrm{ex}}=m_{\mathrm{crit}}$ . No source identifies an originator or date for the conjecture.

Known results

For sufficiently small masses, balls are unique minimizers in the relevant dimensional and exponent ranges, including $n=3$ , $\alpha=1$ .
For $0<\alpha<2$ , minimizers do not exist for sufficiently large masses.
In three dimensions, the explicit ball-splitting threshold is $m_*=5\frac{2^{1/3}-1}{1-2^{-2/3}}\approx3.512$ , but its equality with the existence and ball-minimality thresholds is unproved.

2025 screened-interaction results

A 2025 paper states the Coulomb conjecture explicitly but does not resolve it: its counterexamples concern truncated Coulomb and Yukawa kernels, while the unscreened Coulomb case remains untreated.

Current status (as of September 2026): the conjecture remains open; small-mass ball minimality, large-mass nonexistence, and the explicit threshold are known, but $m_{\operatorname{ball}}=m_{\mathrm{ex}}=m_{\mathrm{crit}}$ is not proved or disproved.

Gamow's conjecture for the Coulomb liquid drop model

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