Invariant surfaces in Euclidean space with a log-linear density

TL;DR

This paper classifies specific invariant surfaces in Euclidean space that satisfy a mean curvature condition influenced by a log-linear density, extending understanding of geometric flows with density functions.

Contribution

It provides a complete classification of $ ext{lambda}$-translating solitons invariant under certain symmetries in Euclidean space.

Findings

Classification of all invariant $ ext{lambda}$-translating solitons

Explicit descriptions of surfaces with translation symmetry

Explicit descriptions of surfaces with rotational symmetry

Abstract

A $\lambda$-translating soliton with density vector $\vec{v}$ is a surface in Euclidean space whose mean curvature $H$ satisfies $2H=2\lambda+\langle N,\vec{v}\rangle$, where $N$ is the Gauss map. We classify all $\lambda$-translating solitons that are invariant by a one-parameter group of translations and a one-parameter group of rotations.