Geometric flows by parallel hypersurfaces

Abstract

In a recent work it was proved that any immersed hypersurface $M^n$ of a space form evolves through the mean curvature flow ($\mathcal{MCF}$) by parallel hypersurfaces if and only if $M^n$ is an isoparametric hypersurface. The goal of this article is to extend the quoted work to immersed hypersurface $M^n$ of ${\mathbb{Q}}_c^n\times \mathbb{R}$ and ${\mathbb{Q}}_c^n\times {\mathbb{S}}^1$ under a geometric condition of the tangential component of $\frac{\partial }{\partial t}$. More exactly, supposing that this tangential component is a principal direction of the second fundamental form, we will show that such a hypersurface evolves through the $\mathcal{MCF}$ by parallel hypersurfaces if and only if $M^n$ is also an isoparametric hypersurface. Moreover, we will prove that any embedded convex hypersurface $M^n$ of a sphere ${\mathbb{S}}^{n+1}$ evolves through the inverse mean curvature flow ($\mathcal{IMCF}$) by parallel hypersurfaces if and only if $M^n$ is an umbilic hypersurface.