This investigation classifies axisymmetric surfaces in Euclidean space, revealing their energy characteristics and implications for stationary surfaces.
We investigate axisymmetric surfaces in Euclidean space that are stationary for the energy E_α=∫_Σ|p|^α\, dΣ. By using a phase plane analysis, we classify these surfaces when they intersect orthogonally the rotation axis. We also give some applications of the maximum principle characterizing the closed stationary surfaces and the compact stationary surfaces with boundary a circle when $α=-2$. Finally, we prove that helicoidal stationary surfaces must be rotational surfaces.
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Dierkes et al. (2025) studied this question.
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