A connection between minimal surfaces and the two-dimensional analogues of a problem of Euler
Year: 2025
Type: article
Abstract: If $$\alpha \in \mathbb R$$ α ∈ R , an $$\alpha $$ α -stationary surface in Euclidean space is a surface $$\Sigma $$ Σ whose mean curvature H satisfies $$H(p)=\alpha |p|^{-2} \langle \nu ,p\rangle $$ H ( p ) = α | p | - 2 ⟨ ν , p ⟩ , $$p\in \Sigma $$ 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 $$\alpha $$ α -stationary surfaces and $$-(\alpha +4)$$ - ( α + 4 ) -stationary surfaces. In particular, there is a correspondence between $$-4$$ - 4 -stationary surfaces and minimal surfaces. Using this duality we give some results of uniqueness of $$-4$$ - 4 -stationary surfaces and we solve the Börling problem.
Author: Rafael López
Institution: Universidad de Granada
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
OpenAlex ID: W4412477003