Translators to Higher Order Mean Curvature Flows in ${\mathbb{R}}^n\times \mathbb{R}$ and ${\mathbb{H}}^n\times \mathbb{R}$

Abstract

We consider translators to the extrinsic flows in ${\mathbb{R}}^n\times \mathbb{R}$ and ${\mathbb{H}}^n\times \mathbb{R}$ (called $r$-mean curvature flows or $r$-MCF, for short) whose velocity functions are the higher order mean curvatures $H_r.$ We show that there exist rotational bowl-type and catenoid-type translators to $r$-MCF in both ${\mathbb{R}}^n\times \mathbb{R}$ and ${\mathbb{H}}^n\times \mathbb{R},$ and also that there exist parabolic and hyperbolic catenoid-type translators to $r$-MCF in ${\mathbb{H}}^n\times \mathbb{R}.$ In addition, we show that there exist Grim Reaper-type translators to Gaussian flow ($n$-MCF) in ${\mathbb{R}}^n\times \mathbb{R}$ and ${\mathbb{H}}^n\times \mathbb{R}$. We also establish the uniqueness of all these translators (together with certain cylinders) among those which are invariant by either rotations or translations (Euclidean, parabolic or hyperbolic). We apply this uniqueness result to classify the translators to $r$-MCF in ${\mathbb{R}}^n\times \mathbb{R}$ and ${\mathbb{H}}^n\times \mathbb{R}$ whose $r$-th mean curvature is constant, as well as those which are isoparametric. Our results extend to the context of $r$-MCF in ${\mathbb{R}}^n\times \mathbb{R}$ and ${\mathbb{H}}^n\times \mathbb{R}$ the existence and uniqueness theorems by Altschuler--Wu (of the bowl soliton) and Clutterbuck--Schnürer--Schulze (of the translating catenoids) in Euclidean space.

45 pages, 28 figures