Rotational self-shrinkers in euclidean spaces

Abstract

The rotational embedded submanifolds of E n +d were first studied by N. Kuiper. The special examples of this type are generalized Beltrami submanifolds and toroidals submanifold. The second author and et. all recently have considered 3 - dimensional rotational embedded submanifolds in 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 E 5 . Moreover, we give the necessary and sufficient conditions for these type of submanifolds to be homothetic solitons for their mean curvature flows.