2025/07/16 by Rafael López, López, Rafael
Mathematics · #35A15 #49Q05 #53A10 #Advanced Differential Equations and Dynamical Systems #Differential Geometry (math.DG) #FOS: Mathematics #Mathematics and Applications #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2507.12371
openalex publication_date 2025/07/16 · openalex created_date 2025/10/14 · openalex updated_date 2026/07/28
If α∈\r, an α-stationary surface in Euclidean space is a surface Σ whose mean curvature H satisfies H(p)=α|p|-2 ⟨ν,p⟩, p∈Σ. These surfaces generalize in dimension two a classical family of curves studied by Euler which are critical points of the moment of inertia of planar curves. In this paper we establish, via inversions, a one-to-one correspondence between α-stationary surfaces and -(α+4)-stationary surfaces. In particular, there is a correspondence between -4-stationary surfaces and minimal surfaces. Using this duality we give some results of uniqueness of -4-stationary surfaces and we solve the Börling problem.