Minimal surfaces in Euclidean space with a log-linear density

Abstract

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

References in corpus (1)

• Subsequent Singularities in Mean-Convex Mean Curvature Flow