Rotational Self-Shrinkers in Euclidean Spaces

Abstract

The rotational embedded submanifold of $\mathbb{E}^{n+d}$ first studied by
 N. Kuiper. The special examples of this type are generalized Beltrami
 submanifolds and toroidals submanifold. The second named authour and at. all
 recently have considered $3-$dimensional rotational embedded submanifolds in 
 $\mathbb{E}^{5}$. They gave some basic curvature properties of this type of
 submaifolds. Self-similar flows emerge as a special solution to the mean
 curvature flow that preserves the shape of the evolving submanifold. In
 this article we consider self-similar submanifolds in Euclidean spaces. We
 obtained some results related with self-shrinking rotational submanifolds in
 Euclidean $5-$space $\mathbb{E}^{5}$. Moreover, we give the necessary and
 sufficient conditions for these type of submanifolds to be homothetic
 solitons for their mean curvature flows.