Compact $\lambda$-translating solitons with boundary

TL;DR

This paper investigates the shape, boundary conditions, and symmetry properties of compact $ ext{lambda}$-translating solitons in Euclidean space, providing existence criteria, area estimates, and symmetry inheritance results.

Contribution

It establishes conditions for the existence of compact $ ext{lambda}$-translating solitons with prescribed boundary curves and analyzes their geometric properties and symmetry inheritance.

Findings

Existence conditions for compact solitons with given boundary curves.

Area estimates related to the height of the soliton.

Symmetry inheritance from boundary to the entire surface.

Abstract

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