3 problems

Let $\Sigma\subset\mathbb{R}^3$ be a $C^{1,1}$-regular axisymmetric surface, with mean curvature $H$ and induced volume form $\mathrm{d}\operatorname{vol}_g$. Dalphin's conjecture.…

Limiting closest-point parameter conjecture. We conjecture that

Axisymmetry conjecture. For any such prescribed area and volume and any constant spontaneous curvature, the minimization problem has a solution, and a minimizer is axisymmetric.