Burcu Bektaş Demirci

In this paper, we study hypersurfaces in the product spaces ${\mathbb{Q}}_{\epsilon }^3\times \mathbb{R}$ for which the tangential component $T$ of the vector field $\frac{\partial }{\partial t}$ is a principal direction, where ${\mathbb{Q}}_{\epsilon }^3$ denotes the three-dimensional non-flat Riemannian space form with sectional curvature $\epsilon =\pm 1$, and $\frac{\partial }{\partial t}$ is the unit vector field tangent to the $\mathbb{R}$-factor. We obtain a local classification of hypersurfaces with three distinct principal curvatures satisfying specific functional relations. Then, we determine the necessary and sufficient conditions for such hypersurfaces to admit an almost Ricci soliton structure with potential vector field $T$. Finally, we prove that the only hypersurfaces admitting such solitons are rotational, by showing that the constructed examples with three distinct principal curvatures do not admit almost Ricci solitons.

In this paper, we study $K^{\alpha }$–translators on parallel surfaces and canal surfaces in 3-dimensional Euclidean space ${\mathbb{E}}^3$. First, we investigate the condition under which two parallel surfaces can become $K^{\alpha }$–translators moving with the same speed $w$. Then, we examine $K^{\alpha }$–translators on canal surfaces and we show that if a canal surface is $K^{\alpha }$–translator, then it must be a surface of revolution in ${\mathbb{E}}^3$. We also provide examples for moving a surface of revolution under $K$–flow (Gauss curvature flow) and $K^{-1/2}$–flow (inverse Gauss curvature flow) along a direction $w=(0,0,1)$ and we illustrate such surfaces using Wolfram Mathematica 10.4. Finally, we prove that no $K^{\alpha }$–translators exist on the parallel surface of a rotational surface obtained from a canal surface with the same speed $w$, while the such rotational surfaces itself is a $K^{\alpha }$–translator with speed $w$.