Compact $λ$-translating solitons with boundary

Abstract

A $λ$-translating soliton with density vector $\vec{v}$ is a surface $Σ$ in Euclidean space ${\mathbb R}^3$ whose mean curvature $H$ satisfies $2H=2λ+\langle N,\vec{v}\rangle$, where $N$ is the Gauss map of $Σ$. In this article we study the shape of a compact $λ$-translating soliton in terms of its boundary. If $Γ$ is a given closed curve, we deduce under what conditions on $λ$ there exists a compact $λ$-translating soliton $Σ$ with boundary $Γ$ and we provide estimates of the surface area in relation with the height of $Σ$. Finally we study the shape of $Σ$ related with the one of $Γ$, in particular, we give conditions that assert that $Σ$ inherits the symmetries of its boundary $Γ$.