4 problems

Let $\Sigma$ be a closed genus-zero surface and let $X:\Sigma\longrightarrow\mathbb{R}^3$ be an axially symmetric immersion with non-constant mean curvature. Let $\mathcal{H}$ deno…

An axially symmetric Helfrich topological sphere is a closed genus-zero axially symmetric surface that is critical for the Helfrich energy. The plane of symmetry is the horizontal…

Let an axially symmetric critical disk be a critical disk for the Helfrich energy with elastic boundary that is invariant under an axial symmetry. Palmer–Pampano's conjecture. Axia…

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.