Two classification results for stationary surfaces of the least moment of inertia

2025/07/16 by Rafael López, López, Rafael · 1 citation
Engineering · #3D Shape Modeling and Analysis #53A10 #Advanced Numerical Analysis Techniques #Advanced Theoretical and Applied Studies in Material Sciences and Geometry #Differential Geometry (math.DG) #FOS: Mathematics #Secondary: 53C42

paper · pdf · doi:10.48550/arxiv.2507.12398

openalex publication_date 2025/07/16 · openalex created_date 2025/10/14 · openalex updated_date 2026/07/28

Abstract

A surface in Euclidean space \r³ is said to be an α-stationary surface if it is a critical point of the energy ∫_Σ|p|^α, where α∈\r. We prove that all ruled α-stationary surfaces are vector planes (for all α) and a type of elongated helicoids (for α=1). The second result of classification asserts that if α\not=-2,-4, any α-stationary surface foliated by circles must be a rotational surface. If α=-4, the surface is the inversion of a plane, a helicoid, a catenoid or an Riemann minimal example. If α=-2, we find many non-spherical cyclic (-2)-stationary surfaces.

Citations

• Stationary surfaces for the moment of inertia with constant Gauss curvature
• Axisymmetric stationary surfaces for the moment of inertia
• Ruled surfaces of generalized self-similar solutions of the mean\n curvature flow
• Invariant surfaces in Euclidean space with a log-linear density
• Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici lattissimo sensu accepti
• Two non existence results for the self-similar equation in Euclidean 3-space

Cited by

• Axisymmetric stationary surfaces for the moment of inertia

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• Axisymmetric stationary surfaces for the moment of inertia
• Stationary surfaces for the moment of inertia with constant Gauss curvature
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• Classification of ruled surfaces in Lorentz-Minkowski space that are stationary for the moment of inertia
• Stationary rotating surfaces in Euclidean space