Axisymmetric solutions of the λ-singular minimal surface equation

Given a unit vector v⃗∈ R³ v → ∈ R 3 and λ ,α ∈ R λ , α ∈ R , a surface Σ ⊂ R³ Σ ⊂ R 3 is called λ λ -singular minimal surface if its mean curvature H satisfies H(p)=α N(p),v⃗ p,v⃗ +λ H ( p ) = α ⟨ N ( p ) , v → ⟩ ⟨ p , v → ⟩ + λ for all p∈ Σ p ∈ Σ , where N is the unit normal to Σ Σ . In this paper, we study the axisymmetric solutions of this equation, identifying two types of surfaces depending on whether the rotation axis is parallel or orthogonal to v⃗ v → . For the first type, we prove the existence of surfaces that orthogonally intersect the rotation axis. For the second type, we provide a complete geometric description.