Minimal surfaces in Euclidean space with a log-linear density

We study surfaces in Euclidean space ${\mathbb R}^3$ that are minimal for a log-linear density $ϕ(x,y,z)=αx+βy+γy$, where $α,β,γ$ are real numbers not all zero. We prove that if a surface is $ϕ$-minimal foliated by circles in parallel planes, then these planes are orthogonal to the vector $(α,β,γ)$ and the surface must be rotational. We also classify all minimal surfaces of translation type.

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