Liquid-drop minimizer conjecture

For $m>0$, define the liquid-drop energy

$$E^{\rm G}(m)=\inf_{|\Omega|=m}\left(\operatorname{Per}(\Omega)+\frac{1}{2}\int_\Omega\int_\Omega\frac{1}{|x-y|}\,\mathrm{d}x\,\mathrm{d}y\right),$$

where $\Omega\subset\mathbb{R}^3$ and

$$m_*=5\frac{2-2^{2/3}}{2^{2/3}-1}.$$

Liquid-drop minimizer conjecture. The energy $E^{\rm G}(m)$ has a minimizer if and only if $m\leqslant m_*$. Moreover, whenever a minimizer exists, it is a ball. The assertion reflects the expected transition from connected spherical nuclei to nonexistence caused by Coulomb fission, and remains open in this stated form.